A NEW APPLICATION FOR 3D-SNAKES

Modelling Electrical Discharges

Gilmario Barbosa dos Santos

University of State of Santa Catarina, UDESC-DCC, Joinville-SC, Brazil

Sidney Pinto da Cunha

Center of Information Technology Renato Archer, CTI-DRVC, Campinas-SP, Brazil

Clesio Luiz Tozzi

University of Campinas, UNICAMP-FEEC-DCA, Campinas-SP, Brazil

Keywords:

B-spline, Snakes, 3D image reconstruction, Camera calibration, Electrical discharges characterization.

Abstract:

A new approach for modelling electrical discharges is proposed. To this purpose, an active contour named 3D-

snake is used that is geometrically represented by a B-spline which evolves in 3D space constrained by internal

and external energies. More speciﬁcally, this external energy come from a pair of images. This new model is

much less dependent on determination of homologous points than the approaches found in the literature for

recovering 3D geometry of electrical discharges. In addition, the proposal discussed here is capable of tracking

the evolution os the electrical discharge taking into account the time dependence between consecutive pairs of

frames in two videos.

1 INTRODUCTION

Computer vision techniques have applied stereopsis

in images of electrical discharges. Some works as

MacAlpine and Qiu (MacAlpine et al., 1999), Qiu et

al. (Qiu and MacAlpine, 2000; Qiu et al., 1999) and

Amarasinghe et al. (Amarasinghe et al., 2007) present

some important results in this ﬁeld but their strategies

are strongly dependent on explicit methods of homol-

ogous determination, and furthermore these methods

are applied for electrical discharges with low level of

curvature and wavy discharges are avoided in their ex-

periments. On the other hand, here we propose a new

approach based on 3D snakes for modelling the longi-

tudinal medial-axes of electrical discharges based on

two digital videos which practically makes correspon-

dences determination unnecessary.

In the bi-dimensional case ((Kass et al., 1988)),

the snakes are modelled by a energy funcional that

is minimized under certain constraints, as a conse-

quence the snake deforms itself looking for some fea-

ture of interest in the image. The 3D snake is a model

based on the same principles, which means that the

3D snake also has a functional of energy with geo-

metric and photometric constraints. The aspect that

differentiates 3D snake is related to its external energy

which is extracted from more than one image, which

means that the external energy must come from 2D

spaces determining a 3D force, i.e., the external en-

ergy emerges from a stereo pair of images.

Although very commonly applied in solving the

functional in bi-dimensional snakes, neither Dynamic

Programming (Amini et al., 1988) nor Greedy Algo-

rithms (Williams and Shah, 1992) are useful in the

case of 3D snakes. The approach here is based on

Ca˜nero (Canero et al., 2000; Canero, 2002) which

does not explore the methods based on meta heuris-

tic.

In the stereopsis of two digital videos, the key as-

pect in recovering a 3D global/world point from im-

ages, is the identiﬁcation of homologous points in the

images. Finding homologous points is a hard problem

in stereopsis, although correlation is usually applied

for this sake but sometimes does not succeed. Consid-

546

Barbosa dos Santos G., Pinto da Cunha S. and Luiz Tozzi C. (2009).

A NEW APPLICATION FOR 3D-SNAKES - Modelling Electrical Discharges.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 546-553

DOI: 10.5220/0001803205460553

 SciTePress

ering a sequence (or video) of stereo pairs of electri-

cal discharge, the 3D-snake approach practically dis-

pense the homologous determination.

The methodology for validating the approach de-

scribed here is based on a set of images built by the

simulation of an image acquisition system that cap-

tures a mathematical curve in evolution in 3D space.

After validation the method was applied on electrical

arcs stereo pairs.

In the next section (2) will be described the pre-

cursor snake model and a the 3D snake model (3).

Following a description of the experiment develpment

is given and the results reported (section 4). Finally,

the approach is applied on images of electrical dis-

charges.

2 THE PRECURSOR SNAKE

MODEL

The model of snakes was ﬁrst proposed by Kass et

al. (Kass et al., 1988), it is a physical inspired model

based on the functional of energy below.

total

int

(v(s)) + E

ext

(v(s))ds. (1)

In the Eq. (1), the snake is the contour v(s) and

coordinates x and y are determined by the parame-

ter s, so v(s) = (x(s), y(s)), and the precursor snake

can be seen as deﬁned in a bi-dimensional cartesian

space. The external energy E

ext

(v(s)) represents the

photometric constraint originated from the image and

should be deﬁned conveniently in order to generate

the force to guide the snake to the object of interest

in the image. In turn, the internal energy E

int

(v(s))

is the geometric constraint representing the smooth-

ness grade (ﬁrst and second order of continuity) of

the active contour. This aspect can be distinguished

by checking the deﬁnition of the internal energy. Note

that, according to Eq. (2) below, the terms

∂v(s)

∂s

and

∂

v(s)

∂s

determine the ﬁrst and second order continuity,

respectively. Also observe that the forces originated

by internal energy are intrinsic characteristics of the

snake.

In Eq.(2), the parameters α and β weigh the terms

in order to control the geometry of the contour deﬁn-

ing how much of it could be wavy or not.

int

(v(s)) =

α(s) ∗



∂v(s)

∂s



+ β(s) ∗



∂

v(s)

∂s



d(s). (2)

In terms of numerical methods the functional is

solved by the relaxation according to equations (3)

and (4) below (Kass et al., 1988), where α and β are

introduced in the pentadiagonal matrix A, described

in (Ihlow and Seiffert, 2005) and (Kass et al., 1988),

called stiffness matrix. The γ parameter is used for

weighing the external forces.

= (A + γI)

−1

t−1

− F

) (3)

= (A + γI)

−1

t−1

− F

) (4)

In fact, Eq. (3) and Eq. (4) can be compressed into

Eq. (5), where v

= (x

, y

) is a point of the snake in bi-

dimensional space and F corresponds to the external

force composed by F

, F

= (A + γI)

−1

t−1

− F) (5)

Although has been initially proposed in 1988 the

active contours is still alive as can be seen in recent

works such as (Thevenaz and Unser, 2008).

3 3D-SNAKE

The 3D-snake is similar to the 2D precursory model

described above but the functional has external ener-

gies deﬁned in 3D space and the external forces act

upon the control points of a B-spline which represents

the 3D-snake geometrically.

The 3D-snake corresponds to a B-spline which de-

forms itself in 3D space in order to match its projec-

tions with a pair of features of interest in two images.

As mentioned before, the external force is recovered

from a pair of images (in fact a pair of vectorial maps)

through triangulation, see Trucco and Verri (Trucco

and Verri, 1998) or another stereo vision method. It

is important to note that for 3D-snakes the external

force does come from the same dimensional space of

the snake itself. This aspect is an important difference

to the prior snake model proposed by Kass et al. (Kass

et al., 1988). Since, in that case the snake evolved in

the same bi-dimensional space from where the exter-

nal forces were extracted.

The 3D-snakes implemented here were inspired

mainly in the works of Ca˜nero (Canero et al., 2000;

Canero, 2002).

3.1 Initialization

Ca˜nero in (Canero et al., 2000; Canero, 2002) de-

scribes a manual determination of homologous from

which a set of 3D points are recovered for initializa-

tion of the ﬁrst 3D-snake. In the proposal described

here, the initialization almost dispense homologous

determination and consists of automatic procedure.

This initialization will be described in details later

on. The next paragraphs are focused in describing the

A NEW APPLICATION FOR 3D-SNAKES - Modelling Electrical Discharges

547

steps needed to get the ﬁrst 3D-snake after determina-

tion of the initial 3D points.

The inaccurate set of 3D points is approximated

(not interpolated) by a third order piecewise polino-

mial B-spline which will represent the 3D-snake. An

interpolation would force the B-spline to go through

all of the 3D data points which is not convenient be-

cause the B-spline resultant will be very wavy. So,

the best choice is doing an approximation in order

to obtain a smooth B-spline which does not neces-

sarily pass through every 3D data points; Rogers and

Adams (Rogers and Adams, 1990) provides practical

methods in order to solve this problem.

The functional of energy associated is deﬁned

and the 3D-snake deforms by minimization of this

functional under geometrical and photometrical con-

straints. This minimization process leads the control

points of the B-spline in order to guide the projections

of the points generated by the B-spline (3D-snake) in

the direction of the features of interest in the pair of

images. When the projections match the features that

have been pointed out, the best conﬁguration of the

3D-snake will have been reached and consequently

the best 3D spatial location of the longilineous object

represented by the snake.

For a clearer understanding, consider a pair of

videos with k pairs of frames obtained in time i ∈

0, 1, ..., k − 1. The proposal is to adjust the 3D snake

, to the pair of vectorial maps (m

i+1

) repre-

senting external forces in time i + 1. So, after time

k− 1 all of the possible conﬁguration of the 3D snake

will be covered and consequently the object repre-

sented by the 3D-snake will be tracked. Since the

cameras have been calibrated it is even possible to

measure this object.

Figure 1: The external force is recovered from the space of

vectorial maps. The resultant 3D vector acts over the nodes

of the 3D-snake (in fact upon the control points of the B-

spline that represents it) deforming it in order to adapt its

own projections over a pair of images.

3.2 The Functional of Energy

The initialization gives the third order B-spline B

it is custumary to describe a B-spline by a matricial

equation like in Eq. (6), where Q represents a point

generated by the B-spline through the set of control

points V and the basis functions matrix N. This ma-

tricial formulation will be used here.

Q = NV (6)

The deformation of the 3D snake is deﬁned by a func-

tional of energy which is very similar to the one pre-

sented in Eq. (1) but here the active contour is repre-

sented by a B-spline:

E(Q) =

int

(Q) + E

ext

(Q)ds. (7)

Similarly to Kass et al. (Kass et al., 1988), the E

int

preserves smoothness at the same time that E

ext

is re-

sponsible for the forces which attract the 3D-snake,

pushing into the features of interest captured in the

images. Again, the minimization occurs by relaxation

such as is deﬁned in Eq. (8), which is similar to Eq.

(5) except that here the minimization acts on the con-

trol points of the B-spline, so that a new set of control

points V

is calculated based on the old set V

t−1

= (H+ γI)

−1

(γV

t−1

− g(Q

t−1

)) (8)

Moreover, other important aspects and some sim-

ilarities between Eq. (8) and Eq. (5) must be empha-

sized, as follows:

• Note, the matrix H corresponds to the stiffness

matrix A as seen in Eq. (5). The construction

of H will be discussed later.

• The vector g(Q) corresponds to the external force

transformed into the space of control points,

therefore, the term V

t−1

− g(Q) attracts the con-

trol points and consequently Q(s) in the direction

of the features of interest depicted in the images.

For calculating g(Q) Ca˜nero (Canero, 2002) rec-

ommends the approximation described in Eq. (9),

where N is the basis functions matrix used in Eq.

(6):

g(Q) ≈ N

ext

(Q) (9)

• The external force F

ext

is recovered from vectorial

maps (from 2D to 3D space). As shown in Fig. 1,

the homologous points q

and q

are associated

to the vectors (A− q

) and (B − q

),respectively

in 2D space (vectorial maps), therefore a triangu-

lation (as in Trucco and Verri (Trucco and Verri,

1998)) of the end points of these vectors (point A

and B) gives the 3D force that acts upon a point in

the B-spline. The set of all 3D forces calculated

builds the vector F

ext

which is transformed into

the space of control points, giving birth to vector

g(Q) in Eq. (9).

By inspecting Fig. 1, geometrically the overall

formula to get F

ext

can be seen in Eq. (10), be-

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

548

low.

ext

(Q) = −∇V(Q) ⇒

ext

(Q) = ϕ

−1

− ∇V

), q

− ∇V

)) − Q

−1

is the operator of triangulation

(10)

Finally, g(Q) is substituted in Eq. (8), in the term

t−1

− g(Q)) which plays the role of attracting the

B-spline’s control points to the features of interest in

images. Since the B-spline represents the 3D-snake,

pushing the ﬁrst implies in guiding the second to the

features of interest.

3.3 The Matrix H

Similarly to matrix A in Eq. (5), matrix H represents

the stiffness of the 3D snake model. The parameters

α and β are embedded in H, so H acts on the control

points of the B-spline that represents the 3D-snake de-

termining its ﬂexibility. Ca˜nero (Canero, 2002) sug-

gests calculating H according Eq. (11), below:

H =

L−1

∑

σ=0

(αP

′

+ βP

′′

(11)

Note that:

• L: the number of spans in the vector of knots of

the B-spline;

• N

: the matrices of span, wich could be calculated

algorithmically,Blake and Isars (Blake and Isard,

2000) present such algorithm;

• G

: the matrices G

present d × N

and are used

to select one subset of control points consecu-

tively. Differently than Blake and Isard (Blake

and Isard, 2000) here the expression in Eq. (12)

will be applied for deﬁning G

, as shown below:

)



1 if, j − b

= i;

0 otherwise.

(12)

Where:

∑

i=0

− d (13)

Note that:

– σ correspondsto one span of the vector of knots

of the B-spline;

– m

is the multiplicity of the i-th knot in the vec-

tor of knots;

– d corresponds to the order of the B-spline.

• P’ and P” According to Ca˜nero (Canero, 2002)

the ﬁrst and second derivatives of P (a Hilbert

matrix

) can be calculated by Eq. (14) and Eq.

Hilbert Matrix H

i−1

j−1

(15). Such formulas differs from (Blake and Is-

ard, 2000) but are more appropriate.

P’ =

(

0 if i = 1 or j = 1;

(i−1)( j−1)

i+ j−3

otherwise

(14)

P” =

(

0 if i < 3 or j < 3

(2−3i+i

)(2−3j+ j

)

i+ j−5

otherwise

(15)

4 VALIDATING THE PROPOSAL

A sequence of image pairs for testing the model were

generated. The images were calculated according to

mathematical function described in the next section

which create an evolving curve in the 3D space. Pro-

vided such images, the 3D-snake should be initialized

and forced to deform by the relaxation as described in

Eq. (8) constrained by the external and internal forces.

Being a model for the real curve, the 3D-snake should

track it. The accuracy of this tracking can be eas-

ily evaluated because the real curve is known. The

methodology for evaluation consists in measuring the

length of each instance of the real curve (L

) for com-

parison with the lengths estimated by the 3d-snake

model (L

snake

). Also, the deviation between real and

estimated lengths is calculated. The curvature of each

real curve is used to evaluate the robustness of the

model for wavy instances of the curve.

Given a point p

the curvature in this point can

be approximated by the second derivative taking into

account its two adjacent neighbors. Eq. (16) below

is used for determination of the total curvature of the

i-th conﬁguration of the curve (Curv

Curv

N−1

∑

j=2

curv

(16)

Note:

curv

=k p

j− 1

− 2∗ p

+ p

j+1

(17)

The percentual deviation of each length measure

of the i-th 3D-snake (L

snake

) is done by Eq. (18) as

shown below:

perc

− L

snake

∗ 100 (18)

4.1 Mathematical Curve

An image database was built from the curves accord-

ing the Eq. ( 19) which describes the family of helixes

depicted in Fig. 2. Each pair of images results from

A NEW APPLICATION FOR 3D-SNAKES - Modelling Electrical Discharges

549

projections of the respective member of this family.

Each helix member is determined by Eq. ( 19) using

(t) =

1000

, w

(t) =

100

and v = 4. The ﬁrst curve is

not exactly an helix but a straight line along the axis

OY that smoothly transforms itself into an helix as an

spring that is strongly stretched and then gradually re-

leased (Fig. 2).

For each incoming value of r

and using the pa-

rameters of the cameras stipulated according to the

geometry shown in Fig. 3 and Fig. 4 the respective

spatial conﬁguration of the curve is captured by the

pair of virtual cameras.

C(t) =







x(t) = r

∗ sen(w

(t) ∗ a);

y(t) = v ∗ a;

z(t) = r

∗ cos(w

(t) ∗ a);

(19)

Notes:

1) a, x, y and z are vectors;

2) a = (θ

, θ

, ..., θ

max

) in radians;

3) w

(t) and w

(t): angular velocities;

4) v: velocity along axis OY;

5) r

: the discretely crescent ray of the helix, r

∈

, ...,t

max

}, t

= 1 and t

j− 1

< t

j+1

Figure 2: (I) Samples of the family of helixes obtained by

Eq. (19). (II) View of curves from ZY plane. (III) And view

from XY plane.

4.2 Images Acquisition System

A sufﬁcient number of points of the mathematical

curve is generated and projected in the respective im-

age plane of two virtual pinhole cameras (see Fig. 4).

Both are deﬁned by their respective extrinsic (R and

T) and intrinsic parameters (focal distance, dimen-

sions of pixels, principal point). In terms of geometry,

the extrinsic parameters describe the relation between

the coordinates of a point in global/world P

and a

point in the 3D camera system P

; here we adopt the

following relation P

= RP

+T and a geometry such

as that shown in Fig. 3.

By repetition of this process, for various spatial

conﬁgurations of the curve, two sets of frames were

consistently created.

Considering the deﬁnition of intrinsic and extrin-

sic parameters for both virtual cameras it is possible

Figure 3: The geometry of a pinhole camera model with

image plane in front of the principal point (focus) of the

camera. Axis OX

, OY

and OZ

deﬁne the global/world

coordinate system, OX

, OY

and OZ

refer to the camera

system.

to project the points generated by the mathematical

function in both image planes, and right cameras (see

Fig. 4). By repetition of this process for various spa-

tial conﬁgurations of the curve two sets of frames are

captured.

Figure 4: A curve in 3D space has its points projected in a

pair of image planes left and right.

4.3 Vectorial Maps

Provided that the two sequences of frames (from the

left and right cameras) were acquired, the next step is

to transform them in vectorial maps for representing

the external forces components.

Usually, image processing operations are done for

emphasizing features of interest in each frame of the

sequence. At the same time, to prepare the images

to be transformed by a differential operator such as

the gradient. In this work the distance transform is

used in order to generate a matrix whose cells rep-

resents a pixel in the respective frame (image) and

stores the distance from this pixel to the projection of

the curve. Then, this map of distances is operated by

the GVF (Xu and Prince, 1998) originating two ma-

trices with the horizontal and vertical components of

the gradient vectors. For the sake of simplicity, these

two matrices will be considered as just one matrix of

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

550

resultant gradient vectors, which will be called as the

vectorial map associated to its respective frame (im-

age). These maps are responsible for the vectors q

and q

B as shown in Fig. 1. Fig. 5 shows the result of

transformation by distance transform and respective

vectorial map.

Three views of the same distance map could be

seen in Fig. 6, emphasizing its topological attributes

with the feature of interest in the bottom of a valley.

Clearly, there are vectors pointing to (or from) this

valley and they can be extracted by applying a differ-

ential operator.

Figure 5: An image after application of distance transform,

this distance map can be transformed by a gradient operator

in order to get a vectorial map (below).

Figure 6: Three views of a distance map. The map looks

like a geographical valley where the feature of interest lying

on its bottom.

4.4 Automatic Initialization

In its ﬁrst ﬂash of existence an electrical discharge

usually resembles a straight line linking two points,

similarly its projections look like low curvature

curves. At this moment the discharge can be seen as

an 3D vector whose extremities are points A and B.

This vector can be easily recovered by triangulation

of two pairs of points projected at the pair of images

taken. Such points corresponds to A

and A

, B

and

in Fig. 7, these are used for triangulation (Trucco

and Verri, 1998) in order to get A and B.

The 3D vector V is deﬁned by V = B − A, and

the set of points C that are generated by this vec-

tor are determined by p = A + cV, where c ∈ R and

0 ≤ c < 1. Geometrically, the set C corresponds to

the reconstruction of the electrical discharge in 3D at

its initial instants of existence. These 3D points are

approximated by a third order B-spline (B

) the ﬁrst

spatial representation of the 3D-snake.

Figure 7: A can be determined by triangulation (Trucco and

Verri, 1998) based on A

and A

its projections, similarly, B

can be found by triangulation by B

and B

.It is possible to

triangulate A

and A

to obtain A, as well as B

and B

obtain B. A and B are extremities of the electrical discharge

in 3D.

4.5 Deformation

The 3D-snake, represented by a B-spline, should de-

form itself according of the minimization of its total

energy functional described in Eq. 1. The snake con-

verges to a stable conﬁguration when a minimum of

energy has been obtained which means that the de-

formation should stop because an equilibrium of the

internal and external forces has been reached. The

goal here is a tracking operation, so the snake needs

to ﬁnd the equilibrium for each pair of vectorial map

available.

Now, consider the set of i pairs of vecto-

rial maps, where m

represents a map in ith

time and associated to the jth camera: M =

{(m

, m

);(m

, m

);...;(m

i−1

, m

i−1

)}. Also, consider

as the 3D-snake obtained for the ﬁrst pair (m

, m

In order to get the next spatial conﬁgurations of the

3D-snake, it is necessary to minimize the functional

in Eq. (1) under the inﬂuence of the next pair of maps.

Following such path, the conﬁguration represented by

results from the evolution of B

adjusted to the pair

, m

), B

results from the evolution of B

adjusted

to (m

, m

) and so on, up to the (i− 1)th pair of maps

when all of the possible spatial conﬁgurations have

been taken in three-dimensional space.

5 RESULTS

The actual lengths of the helixes versus the ones cal-

culated by 3D-snake are shown in Fig. 8. Also, for

each helix the average of its curvature was calculated

and is exhibited in Fig. 9 and the deviation is shown

in Fig. 10.

A NEW APPLICATION FOR 3D-SNAKES - Modelling Electrical Discharges

551

The deviation rises according to the rising of the

curvature of the helixes. On the other hand, the 3D-

snake provides a good estimate for the length of the

electrical discharge, as seen in Fig. 8 the measure-

ments made through the 3D-snake model follow the

proﬁle determined by the actual measurements.

The model works well for medium and low curva-

tures which represents an improvement in comparison

with methods found in the literature .

Figure 8: Real lengthes of the helixes versus lengthes cal-

culated by 3D-snake model.

Figure 9: Real curvature for each helix created by Eq.(19).

5.1 Testing the Proposal with Real

Discharges

Since the approach was validated with sinthetic

curves, it will be applied to real images. A pair of dig-

ital cameras Sony

DSC-P200, at 30 fps, was used

to capture the discharge images produced in the ap-

paratus called Plasma Ball driven by capacitive effect

(Fig. 11). The cameras were calibrated by a chess pat-

tern, according Trucco’s method (Trucco and Verri,

1998). The Plasma Ball is a very safe and cheap way

to produce electrical discharges, it is basically a glass

Figure 10: Average deviation for each measured on 3D-

snake.

bulb ﬁlled with a special gas and a source of alternat-

ing high voltage (not as high as produced by a Tesla

coil). When the glass bulb is touched, from the out-

side, a bright discharge is generated by the electrical

ﬁeld produced on that point. The arcs are produced by

the high voltage and the physical effect called capac-

itive force, in regions inside the ampoule where the

gas become more conductive, see Fig. 12.

Figure 11: Acquisition system: C

(Sony

P100) and C

(Sony

P200) usual digital cameras. The plasma ball is the

apparatus used for producing the electrical discharges.

Figure 12: (I) Details of the Plasma Ball: distance AB is

≈ (60± 0.5)mm. (II) Electrical discharge produced.

A set of image pairs was captured followed by

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

552

the approach proposed to do the measurements of the

discharge, based on the 3D-snake. As an example,

was obtained for the discharge lenghts: ≈ (60.46 ±

0.44)mm. The set of dischargespresents very low cur-

vature, so the distance d

≈ (60 ± 0.5)mm shown in

(Fig. 12) is a good result for the true length, allowing

characterization of the electrical discharges, such as

current density and other electrical parameters.

After these experiments the method will be ap-

plied to high voltage transmission lines.

6 CONCLUSIONS

This work described and validated an approach to be

applied in modelling of electrical discharges captured

in a sequence of stereo pairs. The approach was tested

with an image database built by a consistent strategy

and the cameras were based on the classical theoreti-

cal pinhole camera model.

The results obtained by 3D-snake in estimating

the length of the curves are coherent with the real

lengths. Since the cameras are calibrated it is also

possible to determinate the real position of the electri-

cal discharge during the time of the image acquisition,

so the approach proposed here can work as a strategy

for tracking. A new ﬁeld of application for 3D active

contours is opened, such as the tracking of electrical

discharges captured in a pair of digital videos and the

study of fast events.

Thus, in the near future this methodology will

be applied in the studies of real electrical discharges

where, certainly, will be found new constraints and

more critical requirements to be evaluated.

ACKNOWLEDGEMENTS

The authors thank the ﬁnancial support received from

FAPESP - The State of S˜ao Paulo Research, from

CNPq - The National Council for Scientiﬁc and Tech-

nological Development and from CAPES - Coordina-

tion of Improvement of Higher Level Education Per-

sonnel. Special thanks to Mr. Marco Iacovacci.

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