COMPARATIVE STUDY OF CONTOUR FITTING METHODS IN

SPECKLED IMAGES

Mar´ıa E. Buemi, Juliana Gambini, Julio C. Jacobo, Marta E. Mejail

Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Computaci´on

Ciudad Universitaria, Pabell´on I, C1428EGA

Ciudad Aut´onoma de Buenos Aires – Rep´ublica Argentina

Alejandro C. Frery

Universidade Federal de Alagoas, Departamento de Tecnologia da Informac¸˜ao

BR 104 Norte km 97, 57072-970 Macei´o, AL – Brazil

Keywords:

Speckle, contour ﬁtting, B-Spline, anisotropic diffusion, maximum likelihood.

Abstract:

Images obtained with the use of coherent illumination are affected by a noise called speckle, which is inherent

to this type of imaging systems. In this work, speckled data have been statistically treated with a multiplicative

model using the family of G distributions. One of the parameters of these distributions can be used to charac-

terize the different degrees of roughness found in speckled data. We used this information to ﬁnd boundaries

between different regions within the image.

Two different region contour detection methods for speckled imagery, are presented and compared. The ﬁrst

one maximizes a likelihood function over the speckled data and the second one uses anisotropic difussion over

roughness estimates. To represent detected contours, the B-Spline curve representation is used.

In order to compare the behaviour of the two methods we performed a Monte Carlo experience. It consisted

of the generation of a set of test images with a randomly shaped region, which is considered in the literature

as a difﬁcult contour to ﬁt. Then, the mean square error was calculated for each test image, for both methods.

1 INTRODUCTION

Several types of imaging devices employ coherent il-

lumination as, for instance, Synthetic Aperture Radar

(SAR), sonar, laser and ultrasound-B. The images

generated by these devices are affected by a noise

called speckle, a kind of degradation that does not

obey the classical hypotheses of being Gaussian and

additive. Speckle noise reduces the ability to extract

information from the data, so specialized techniques

are required to deal with such imagery.

Speckled data have been statistically modeled un-

der the multiplicative model using the family of G

distributions, since these probability laws are able to

describe the observed data better than other laws, spe-

cially in the case of rough and extremely rough areas.

As a case of interest, in SAR images such sit-

uations are common when scanning urban spots or

forests on undulated relief, and for them the more

classical Γ and K distributions do no exhibit good

performance (Frery et al., 1997; Mejail et al., 2001).

Under the G model, regions with different degrees

of roughness can be characterized by the parameters.

Therefore, this information can be used to ﬁnd bound-

aries between regions with different textures.

An example is shown in Figure 1, where the

dashed lines show the ideal boundary and the solid

lines presents typical speckled data set associated to

this edge in semilogarithmic scale. As can be seen

in this ﬁgure, edge detection in speckled imagery is a

tough task due to the low signal-to-noise ratio.

0 20 40 60 80 100

0.1 0.5 1.0 5.0 10.0 50.0

Position

Boundary and Data

Figure 1: Edge (dashed lines) and speckled data (solid

lines).

309

E. Buemi M., Gambini J., C. Jacobo J., E. Mejail M. and C. Frery A. (2007).

COMPARATIVE STUDY OF CONTOUR FITTING METHODS IN SPECKLED IMAGES.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 309-316

 SciTePress

On the other hand, contours formulated by means

of B-Splines present several advantages: allow local

control, have local representation, require few para-

meters and are intrinsically smooth.

We compare two strategies for boundary detection

with B-Spline deformable contours: one that maxi-

mizes a likelihood function that directly employs the

speckled image values that obey a G

law (Gambini

et al., 2006) and another strategy that uses anisotropic

diffusion over roughness estimates based on this sta-

tistical distribution (Gambini et al., 2004).

In order to compare the behavior of both methods

we performed a Monte Carlo experience. It consists

of the generation of a set of test images with a ran-

domly shaped region. For 2D shapes of ﬁxed perime-

ter, (Hero et al., 1999) established that disk-shaped

objects are the easiest to estimate, while ﬂower-

shaped objects are the hardest to estimate, among the

class of objects representable by the B-Spline basis.

Then, we generate random samples obeying the G

distribution with different parameters for the points

inside and outside the ﬂower contour. Finally, we cal-

culate the error ﬁtting. We conclude that the maxi-

mum likelihood model assuming the G

distribution

for the speckled data is the best of these two edge de-

tection procedures with respect to both the error and

the computational cost.

The structure of this paper is as follows: section 2

describes the statistical model used for single chan-

nel speckled data, section 3 provides a brief account

of B-spline curve ﬁtting, section 4 describes the al-

gorithms in detail, section 5 presents the error evalu-

ation methodology, section 6 presents the results and

section 7 concludes the paper.

2 THE G DISTRIBUTION FOR

SPECKLED DATA

Speckled images can be modeled as the product of

two independent random ﬁelds, one corresponding

to the backscatter X and other corresponding to the

speckle noise Y (Goodman, 1976):

Z = X ·Y. (1)

For amplitude data, the speckle noiseY is modeled

as a Γ

−1/2

(n,n) distributed random variable, where n

is the number of looks used to generate the image; this

parameter is known or estimated beforehand, and it is

valid for the whole image.

The most general model for the backscatter X here

considered is the Generalized Inverse Gaussian law

(Barndorff-Nielsen and Blaesild, 1983; Jorgensen,

1982; Seshadri, 1993), denoted as N

−1/2

(α,λ,γ).

For particular values of the parameters of the N

−1/2

distribution, the Γ

1/2

(α,λ), and the Γ

−1/2

(α,γ) distri-

butions are obtained. These, in turn, give rise to the

and the G

distributions for the return Z, respec-

tively. See(Mejail et al., 2001).

The G

distribution represents an attractive choice

for SAR data modeling, given its tractability, expres-

siveness and capability of retrieving detailed informa-

tion from the data (Quartulli and Datcu, 2004). Its

density function for amplitude data, is given by

(z) =

(n− α)

Γ(−α)Γ(n)

2n−1

(γ+ z

n−α

−α,γ,z > 0, n ≥ 1. (2)

This situation is denoted Z ∼ G

(α,γ,n), being its

moments

) =





r/2

Γ(−α+ r/2)

Γ(−α)

Γ(n+ r/2)

Γ(n)

, (3)

if −α > r or inﬁnite otherwise.

Speckled data is described in this paper by the G

law. Given the data, the statistical parameters are es-

timated and this information is used to extract region

boundaries present in the image.

2.1 Parameter Estimation

As presented in equation (2), the parameter α of the

distribution is deﬁned for negative values. Esti-

mation is crucial in many applications and, besides

that, the value of this parameter is immediately in-

terpretable in terms of target roughness; this inter-

pretability will be treated in detail in section 2.2.

In this work sample moments parameter estima-

tion method (MO for short) is used. This technique

is based on replacing theoretical moments by sample

observations, and then calculating the unknown para-

meters.

To estimate α and γ it is necessary to estimate two

moments. In this work moments of order 1/2 and 1,

namely m

1/2

and m

respectively, will be used. From

equation (3), these moments are given by

1/2





1/4

Γ(−α− 1/4)Γ(n + 1/4)

Γ(−α)Γ(n)

, (4)





1/2

Γ(−α− 1/2)Γ(n + 1/2)

Γ(−α)Γ(n)

, (5)

for α < −1/4 and α < −1/2, respectively. Then, us-

ing equations (4) and (5),

α can be determined as the

solution of

α) − ζ = 0, (6)

where

α) =



−

α−



Γ(−

α)Γ



−

α−



(7)

and

ζ =

1/2

Γ(n)Γ(n+

)

(n+

)

, (8)

and then substituting the value of

α in equation (4)

or in equation (5) the value of

γ is found. It can be

noticed that g(

α) converges asymptotically to one as

α → −∞. As ζ is a random variable that can take val-

ues greater than one, there are cases for which equa-

tion (6) does not have a solution. The lower the value

of the α parameter, the higher the probability that a

solution for equation (6) does not exist. This issue

was solved by (Mejail et al., 2003) replacing unob-

served estimates by the median of surrounding values.

2.2 Parameter Interpretation

One of the most important features of the G

distrib-

ution is that the estimated values of the parameter α

have immediate interpretation in terms of roughness.

For values of α near zero, the imaged area presents

very heterogeneous gray values, as is the case of ur-

ban areas in SAR images. As we move to less hetero-

geneous areas like forests, the value of α diminishes,

reaching its lowest values for homogeneous areas like

pastures and certain types of crops. This is the reason

why this parameter is regarded to as a roughness or

texture measure.

3 B-SPLINE REPRESENTATION

We use the B-Spline curve representation for describ-

ing object contours in a scene. In the following, a

brief review of B-Spline representation of contours

is presented; for more details see (Blake and Isard,

1998) and (Rogers and Adams, 1990).

Let {Q

,...,Q

−1

} be a set of control points,

where Q

= (x

)

∈ R

, 0 ≤ n ≤ N

− 1, and let

< s

< · · · < s

L−1

} ⊂ R be a set of L knots.

A B-Spline curve of order d is deﬁned as a weighted

sum of N

polynomial basis functions B

n,d

(s) of de-

gree d − 1, within each interval [s

i+1

] with 0 ≤

i ≤ L− 1. The spline function is r(s) = (x(s),y(s))

0 ≤ s ≤ L− 1, being

r(s) =

−1

∑

n=0

n,d

(s)Q

, (9)

and

x(s) = B

(s)Q

(10)

y(s) = B

(s)Q

(11)

where the basis functions vector B(s) of N

compo-

nents is given by B(s) = (B

0,d

(s),...,B

−1,d

(s))

The weight vectors Q

and Q

give the ﬁrst and sec-

ond components of Q

, respectively.

The curves used in this work are closed, with

d = 3 or d = 4, and are speciﬁed by periodic B-Spline

basis functions.

4 BOUNDARY DETECTION

In this section we present the methods developed to

detect region boundaries in speckled imagery.

Let E be a scene made up by a background B and a

set of k distinct regions {R

,...,R

} with bound-

aries {∂R

,...,∂R

}, respectively. Each of these re-

gions and the background are considered to be ran-

dom ﬁelds of independent, identically distributed,

random variables obeying the G

distribution and

characterized by the values of their statistical para-

meters (α

,γ

), 1 ≤ h ≤ k. The only assumption we

make is that regions and background have different

textures, i.e., if α

is the background roughness then

6= α

for every 1 ≤ h ≤ k.

For each region R

, we want to ﬁnd the curve C

that ﬁts the boundary ∂R

in the image. As shown

in section 2.2, there is a correspondence between the

various areas present in the image and the parameters

of roughness and scale. From this correspondence a

classiﬁcation of the image can be done.

The algorithms we propose aim at separating re-

gions of a certain speciﬁed type from the rest of the

image, so we obtain a ﬁrst rough approximation, or

seed: the starting regions of interest. The contour ex-

traction algorithms work over these regions instead of

analyzing the whole image reducing, thus, the com-

putational effort.

This initial region detection is computed over non-

overlapping blocks of the image using as input the

number of blocks and the type of region to be detected

(homogeneous,heterogeneous or extremely heteroge-

neous); the details are presented in (Gambini et al.,

2006). Once k regions are identiﬁed, each centroid

, 1 ≤ h ≤ k, is computed.

If a point belongs to the object boundary, then a

sample taken from its neighborhood should exhibit

a change in its properties and, therefore, could be

considered a transition point. In order to ﬁnd tran-

sition points, N segments s

(i)

, 1 ≤ i ≤ N, of the form

(i)

c p

are considered for each region, being c the

centroid of the initial region, the extreme p

a point

outside of the region. It is necessary for the centroid

c to be in the interior of the object whose contour is

sought. A strip S

(i)

is deﬁned over each segment s

(i)

This procedure is illustrated in the ﬂuxogram of Fig-

ure 2 and in Figures 3 and 4 with a SAR image.

Figure 2: Sequence of operations leading from the data con-

ditioned on the blocks (Z | B) to the n strips S

(i)

, 0 ≤ i ≤

n−1, for each region 1 ≤ h ≤ k ≤ i ≤ n−1, for each region

1 ≤ h ≤ k.

Figure 3: Initial region R

with the segments s

(i)

, extreme

point p

and center c

on a SAR image.

In this work we present a comparative study of two

new methods for contour detection in speckled im-

agery using statistical properties of the data and B-

Spline curve representation. One of them uses Maxi-

mum Likelihood over speckled data and the other uses

Anisotropic Diffusion over the roughness estimates.

Figure 4: Strip S

(i+1)

deﬁned over the s

(i+1)

on a SAR im-

age.

As we explain in the following section.

4.1 Maximum Likelihood (ML)

The general problem consists of ﬁnding a transition

point within each strip S

(i)

. These points will be

sought using the data along the discrete version of

(i)

, for which we will not introduce a new notation

(see Figure 2):

(i)



(i)

,...,z

(i)



, 1 ≤ i ≤ N. (12)

For each segment s

(i)

, 1 ≤ i ≤ N, we consider the fol-

lowing partition

(i)

∼ G

(α

,γ

), k = 1, . .., j (13)

(i)

∼ G

(α

,γ

), k = j + 1,...,m (14)

where for each k, with 1 ≤ k ≤ m, z

(i)

is the real-

ization of the random variable Z

(i)

. The parameters

(α

,γ

) and (α

,γ

) characterize the region and its

background, respectively.

In order to ﬁnd the transition point on each seg-

ment s

(i)

, an objective functionis considered: the like-

lihood of the sample which is given by

L( j) =

∏

i=1

Pr(z

;α

,γ

) ·

∏

i= j+ 1

Pr(z

;α

,γ

). (15)

Alternatively, we can maximize

ℓ( j) = lnL( j) =

∑

i=1

ln( f

;α

,γ

))

∑

i= j+ 1

ln( f

;α

,γ

)). (16)

According to equation (2)

ℓ( j) =

∑

i=1



Γ(n−α

2n−1

Γ(−α

)Γ(n)

(

+nz

)

n−α



∑

i= j+ 1



Γ(n−α

2n−1

Γ(−α

)Γ(n)

(

+nz

)

n−α



. (17)

Finally, the estimated index on the segment that

corresponds to the transition point

j is given by

j = argmax

ℓ( j). (18)

The scheme of this procedure is shown in Algo-

rithm 1.

Algorithm 1 Edge detection by maximum likelihood

using raw data.

1: for each segment s

(i)

, i = 1,... , N do

2: Estimate the parameters (α

,γ

) and (α

,γ

3: Find the index

j on the segment s

(i)

that maxi-

mizes equation (18); it corresponds to the bor-

der point b

in the image.

4: end for

5: Build the B-Spline curve that interpolates the

boundary points {b

,...,b

In Figure 5 the result of applying algorithm 1 to a

real SAR image, is shown.

Figure 5: Maximum likelihood edge detection in a SAR

image of Munich.

4.2 Anisotropic Diffusion (AD)

Another way of ﬁnding the transition point on a line

(i)

is to estimate the α parameter in a window cen-

tered on each pixel on the line using the data on the

strip S

(i)

. Then an array

(i)

= [

(i)

,...,

(i)

] of esti-

mates of the α parameter is obtained. If a point lies

on the boundary between two regions, then it exhibits

an abrupt change in the values of the α estimates.

(Perona and Malik, 1990) proposed an algorithm

that combats noise preserving boundary features. In

its continuous version, it consists of producing a se-

quence of images I(·,·,t), t ≥ 0, according to the fol-

lowing equation:

∂I(x,y,t)

∂t

= ∇· [g(k∇Ik)∇I], (19)

where I(·,·,0): R

→ R

is the original image, t is

an artiﬁcial time parameter, ∇I is the image gradient,

k∇Ik is the image gradient magnitude, ‘∇ · ’ denotes

the divergence, g: R → [0, 1] is an edge detection

function with the only constraints that (i) g(x) → 0

monotonically when x → ∞, and (ii) g(x) → 1 when

x → 0. More details can be found in, among others,

(Weickert, 1998).

The position of the discontinuity on s

(i)

is found

by convolving the smoothed roughness estimates with

a cyclic border detection operator, followed by a con-

venient thresholding. The scheme of this procedure is

shown in Algorithm 2.

Algorithm 2 Edge detection by anisotropic diffusion.

1: for each segment s

(i)

, i = 1,... , N do

2: Estimate the parameter α for each pixel on s

(i)

using a sliding window. This generates an ar-

ray

(i)

= [

(i)

,...,

(i)

] of estimated values of

α.

3: Smooth the array

(i)

using anisotropic diffu-

sion. This generates the smoothed estimates

array

(i)

4: Find b

, the position on line s

(i)

that corre-

sponds to the maximum discontinuity among

the values in the smoothed array

(i)

convolv-

ing with the mask [−2,−1, 0,1,2].

5: end for

6: Build the B-Spline curve that interpolates the

boundary points {b

,...,b

In Figure 6, the results of applying Algorithm 2,

are shown. Having proposed two edge detection tech-

niques, we now proceed to their comparative assess-

ment.

5 CONTOUR FITTING ERROR

This section is devoted to the study of the error com-

mitted in the determination of contours by applying

the segmentation methods described above to a series

of randomly generated images.

Figure 6: Edge detection by anisotropic detection in a SAR

image of Munich.

The error evaluation algorithm is applied to a fam-

ily of synthetic random images of ﬂower shaped re-

gions {F

}

k=1,...,n

. Then, the error of approximating

these regions by the obtained B-Spline curves is es-

timated. The {F

}

k=1,...,n

are formed in two stages:

ﬁrst a random region is generated, and then the back-

ground and foreground are simulated.

The random region boundary is generated accord-

ing to a parametric curve in polar coordinates given

f(s;η,β,δ) = (θ(s),ρ(s;η,β,δ)), s ∈ [0,S]

θ(s) = s

2π

ρ(s;η,β,δ) = η− δcos(βθ(s)),

(20)

where η is the ﬂower radius, β is the number of petals

and 2δ is the petal depth.

In this work the parameters η, β and δ are considered

to be independent random variables uniformly distrib-

uted the sets [5,20], {15,...,50} and [2, 10], respec-

tively.

After a region boundary is generated, G

dis-

tributed speckle noise is added to the image. Fig-

ure 7shows some of these simulated images, along

with the estimated boundaries.

In order to calculate the error in boundary ﬁtting

we consider ∂R to be the boundary of the region to be

segmented and C the resulting curve. Let s

,...,s

a set of radial lines given by s

= λ~u

+ c j = 1...m,

where ~u

is a unitary vector that determines the line

direction, and c is the centroid of region R. The point

c and the vectors ~u

j = 1,...,m are the same for all

of the test images. Let

be the intersection points

between curve C and line s

, and let

be the inter-

section points between curve ∂R and the line s

∩ C =

, s

∩ ∂R =

. (21)

(a) (b) (c)

(d) (e) (f)

Figure 7: Synthetic images and detected boundaries with

maximum likelihood method: (a) Image F

with η = 10,

β = 45, δ = 6, (b) Boundary curve F

, (c) Image F

with

η = 7, β = 46, δ = 5, (d) Boundary curve F

, (e) Image F

with η = 9, β = 48, δ = 5 and (f) Boundary curve F

The distance between C and ∂R can be then deﬁned

d(∂R,C) =

∑

j= 1



−



, (22)

where N is the number of segments. This is a measure

of the error committed when estimating ∂R by C .

Algorithm 3 shows the procedure for calculating

the error.

Algorithm 3 Contour Fitting Error.

1: Generate a set of test images using equation (20)

and the G

distribution.

2: for each method do

3: for each image do

4: Find the ﬁtting curve for the ﬂower contour

through the method in evaluation.

5: for j = 0,...,N do

6: Find the points

and

, using equa-

tion (21).

7: Find the distance as in equation (22).

8: end for

9: Return d(∂R,C).

10: end for

11: end for

With the error values calculated in Algorithm 3 for

each of the test images, a graphic that serves as an aid

for the comprehension of each method and that allows

for comparisons between the two methods, is done.

Let e

max

be the maximum error incured when ap-

plying this method. Let us consider the error interval

[0,e

max

] and let partition it as {e

,...,e

}, so

= e

+ ∆ ∗ i, i = 0,...,m (23)

where m is such that e

max

= e

+ ∆ ∗ m and ∆ is the

partition length. We deﬁne the histogram function h :

[0,e

max

] → N such that

h(e

) = #{F

≤ d(∂R

) < e

i+1

}, i = 0,...,m−1.

(24)

We call h

the function h calculated after applying

the method M. We say that a method is efﬁcient if the

valuesof h are high for values close to zero. We deﬁne

function h

: [0,e

max

] → N, by

h(e

) = #{F

: d(∂R,C) ≤ e

}, i = 0,...,m− 1, (25)

We call h

the accumulated histogram function

calculated for method M. A method M is more efﬁ-

cient than another method K if, for a given error value

, the condition

) > h

) (26)

holds.

A Monte Carlo experience was conducted in or-

der to assess the error committed by our proposal, us-

ing 108 simulated images according to the aforemen-

tioned model.

6 RESULTS

In this section, the error committed in the Maximum

Likelihood and Anisotropic Diffusion methods, is es-

timated. A Monte Carlo experience was conducted

in order to assess the error committed by our pro-

posal. We generated 108 simulated images with data

obeying the G

(α,1,1) distribution, with parameters

α = −3 for the ﬂower and α = −10 for the back-

ground. Then, the contour ﬁtting error is calculated

using Algorithm 3, as explained in section 5.

The result for each method is an array of values

corresponding to the error committed in each test im-

age. Then, functions h

) and h

), i = 0, ... , m

using equations (24) and (25), respectively, are calcu-

lated.

Figures 8 and 9 show the error histograms h

)

for the methods ML and AD, respectively. In both

graphics, the horizontal axis corresponds to the error

intervals, and the vertical axis indicates the number of

images with error within each interval.

In order to visualize the comparison between the

errors for both methods, Figure 10 depicting their er-

ror histograms, is shown.

The error values shown here are between 0.0 and

1.0 because the values near zero are of more inter-

est. It is observed that, for the ML method, most of

Figure 8: Error histogram for the ML method.

Figure 9: Error histogram for the AD method.

the images have error values less than 0.6, while for

the AD method, most of the images have error values

greater than 1.0. In section 7, conclusions based on

these graphics are derived about the behavior of both

methods.

In Figure 11 the accumulated error h

) , given

by equation 25, is shown for both methods. In this

graphic it can be seen that there are 81 images with

error below 1.0 for the ML method, while for the AD

method there are only 26 images in that condition. It

can also be seen that

(e) > h

(e) (27)

for e < 1.7.

7 CONCLUSIONS

The goal of this paper was to compare two contour

detection methods for speckled data: Maximum Like-

lihood, performed over the raw data, and Anistropic

Diffusion, over estimates of the α parameter of the

Figure 10: Error histograms for each method: the contin-

uous line with bullets ’•’ corresponds to the ML method,

and the dashed line with the triangles ’△’ corresponds to

the AD method.

Figure 11: Accumulated errors for both methods: continu-

ous line ML method, dashed line AD method. The horizon-

tal line indicates 80% level.

distribution. To this end, a Monte Carlo expe-

rience was conducted in which a region pattern was

randomly generated and then it was submitted to both

boundary ﬁtting algorithms. The error values for the

methods under study was then calculated.

In Figure 11 we observe that in the case of

the Maximum Likelihood method, 80% of the seg-

mented images have errors below 1.1, and that for the

Anisotropic Diffusion method, 80% of the segmented

images have errors below 1.8. This indicates that the

ﬁrst boundary ﬁtting method is better than the second.

Regarding the computational cost, it was also ob-

served that Maximum Likelihood was signiﬁcantly

faster than Anisotropic Diffusion.

REFERENCES

Barndorff-Nielsen, O. E. and Blaesild, P. (1983). Hiperbolic

Distributions. Kotz and Johnson eds., Encyclopedia of

Statistical Science, Wiley, 3:700–707.

Blake, A. and Isard, M. (1998). Active Contours. Springer

Verlag.

Frery, A. C., M¨uller, H.-J., Yanasse, C. C. F., and

Sant’Anna, S. J. S. (1997). A model for extremely het-

erogeneous clutter. IEEE Transactions on Geoscience

and Remote Sensing, 35(3):648–659.

Gambini, J., Mejail, M., Berlls, J. J., and Frery, A. (2006).

Feature extraction in speckled imagery using dynamic

b-spline deformable contours under the G

models.

IJRS International Journal of Remote Sensing. to ap-

pear.

Gambini, M. J., Mejail, M., Jacobo-Berlles, J. C., Muller,

H., and Frery, A. C. (2004). Automatic contour detec-

tion in SAR images. In Proceedings EUSAR04.

Goodman, J. W. (1976). Some Fundamental Properties of

Speckle. Journal of the Optical Society of America,

66:1145–1150.

Hero, A. O., Piramuthu, R., Fessler, J. A., and Titus, S. R.

(1999). Minimax emission computed tomography us-

ing high-resolution anatomical side information and

b-spline models. IEEE Transactions on Information

Theory, 45(3):920–938.

Jorgensen, B. (1982). Statistical Properties of the General-

ized Inverse Gaussian Distribution. Lecture Notes in

Statistic, Springer-Verlag, 9.

Mejail, M., Jacobo-Berlles, J. C., Frery, A. C., and Bustos,

O. H. (2003). Classiﬁcation of SAR images using a

general and tractable multiplicative model. Interna-

tional Journal of Remote Sensing, 24(18):3565–3582.

Mejail, M. E., Frery, A. C., Jacobo-Berlles, J., and Bus-

tos, O. H. (2001). Approximation of distributions for

SAR images: Proposal, evaluation and practical con-

sequences. Latin American Applied Research, 31:83–

92.

Perona, P. and Malik, J. (1990). Scale-space and edge detec-

tion using anisotropic diffusion. IEEE Trans. Pattern

Anal. Machine Intell., 12(7):629–639.

Quartulli, M. and Datcu, M. (2004). Stochastic geomet-

rical modelling for built-up area understanding from

a single SAR intensity image with meter resolution.

IEEE Transactions on Geoscience and Remote Sens-

ing, 42(9):1996–2003.

Rogers, D. F. and Adams, J. A. (1990). Mathematical El-

ements for Computer Graphics. McGraw-Hill, New

York, USA, 2 edition.

Seshadri, V. (1993). The Inverse Gaussian Distribution:

A Case of Study in Exponential Families. Claredon

Press, Oxford, ﬁrst edition.

Weickert, J. (1998). Anisotropic Diffusion in Image

Processing. Teubner-Verlag, ﬁrst edition.