REAL-TIME ELLIPSE FITTING, 3D SPHERICAL OBJECT

LOCALIZATION, AND TRACKING FOR THE ICUB SIMULATOR

Nicola Greggio

1,2

, Alexandre Bernardino

, Cecilia Laschi

, Paolo Dario

and Jos´e Santos-Victor

ARTS Lab - Scuola Superiore S.Anna, Polo S.Anna Valdera, Viale R. Piaggio 34, 56025 Pontedera, Italy

Instituto de Sistemas e Rob´otica, Instituto Superior T´ecnico, 1049-001 Lisboa, Portugal

Keywords:

Humanoid robotics, Machine vision, Pattern recognition, Least-square ﬁtting, Algebraic distance.

Abstract:

This paper presents the implementation of real-time tracking algorithm for following and evaluating the 3D

position of a generic spatial object. The key issue of our approach is the development of a new algorithm

for pattern recognition in machine vision, the Least Constrained Square-Fitting of Ellipses (LCSE), which

improves the state of theart ellipse ﬁtting procedures. It is a robust and direct method for the least-square ﬁtting

of ellipses to scattered data. Although it has been ellipse-speciﬁcally developed, our algorithm demonstrates to

be well suitable for the real-time tracking any spherical object, and it presents also robustness against noise. In

this work we applied it to the RobotCub humanoid robotics platform simulator. We compared its performance

with the Hough Transform and with its original formulation, made by Fitzgibbon et Al. in 1999, in terms of

robustness (success/failure in the object detection) and ﬁtting precision. We performed several tests to prove

the robustness of the algorithm within the overall system. Finally we present our results.

1 INTRODUCTION

The impressive advance of research and development

in robotics and autonomous systems over the past few

years has led to the development of robotic platforms

of increasing motor, perceptual, and cognitive capa-

bilities. These achievements are opening the way for

new application opportunities that will require these

systems to interact with other robots or nontechni-

cal users during extended periods of time. The ﬁnal

goal is creating autonomous machines that learn how

to execute complex tasks and improve their perfor-

mance throughout their lifetime. Motivated by this

objective the RobotCub (ROBotic Open-Architecture

Technology for Cognition, Understanding and Behav-

ior) project has been developed (Sandini et al., 2007).

This is a research initiative dedicated to the realization

of embodied cognitive systems.

1.1 Related Work

The detection of circular objects is fundamental

in many applications, other than the developmental

RobotCub scenarios. An example is in the rescue em-

ployment of robotics platforms. Common situations

that employ rescue robots are mining accidents, ur-

ban disasters, hostage situations, and explosions. Cur-

rently, the research in rescue robotics is very fruitful

(Carpin et al., 2007). In addition, the NIST imple-

mented a simulator, USARSim (Urban Search And

Rescue Simulation) in order to develop rescue robots

(Wang et al., 2003). A clear and precise object recog-

nition is fundamental for such robots to ﬁnd an acci-

dent victim as soon as possible with the highest pre-

cision as possible. Circular ad elliptical objects can

occur in body parts, such as head, and eyes. Vamossy

et Al. applied an ellipse detection algorithm to a res-

cue robot (Vamossy et al., 2003) in 2003, while, more

recently, Greggio et Al. used an ellipse detection al-

gorithm for recognizing the ball within the RoboCup

context (Greggio et al., 2009).

Other work in the robotics implementation of el-

lipse pattern recognition techniques has been per-

formed. Deniz et Al. used an ellipse detection al-

gorithm for face detection (Deniz et al., 2002). In

their work the authors focussed more on Human-

computer interaction. Moreover, Vincze et Al. used

a RANSAC-like method to ﬁnd ellipses in real-world

examples (Vincze et al., 2000). They made exper-

iments to validate the capabilities of the approach

with in real contexts. Finally, Teutsch et Al. ap-

plied the ellipse recognition in industrial processes,

focussing on the real-time characteristics of their ap-

proach (Teutsch et al., 2006).

248

Greggio N., Bernardino A., Laschi C., Dario P. and Santos-Victor J..

REAL-TIME ELLIPSE FITTING, 3D SPHERICAL OBJECT LOCALIZATION, AND TRACKING FOR THE ICUB SIMULATOR.

DOI: 10.5220/0003543502480256

In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 248-256

ISBN: 978-989-8425-75-1

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

1.2 Our Contribution

In this paper we implemented for the ﬁrst time in a

real context our least-square ﬁtting of ellipses tech-

nique (Greggio et al., 2010). We tested our new al-

gorithm, the B2AC (Fitzgibbon et al., 1999), and the

Hough transform (Leavers, 1992) under the same ex-

perimental conditions. We choose an actual task, i.e.

3D localization of a ball, and we compared these al-

gorithms’ performance in terms of overall localiza-

tion precision, and their robustness in terms of suc-

cess/failure detection of the object. We used the

simulation of a state of art robotics platform, the

RobotCub, in order to test it at best before doing this

with the real platform.

1.3 Outline

This paper is organized as follows. In sec. 2 we

will discuss the state of the art problem of the least-

square ﬁtting of ellipses, the original algorithm and

its drawbacks with some solutions proposed in the re-

cent years. In sec. 3 we will propose our approach for

sidestepping the original numerical instability prob-

lem. Then, in sec. 4 we will describe the RobotCub

robotics platform, in terms of its mechanics and the

simulator we used. Furthermore, in section 5 we will

brieﬂy explore our vision algorithms. In sec. 6 we

will describe our experimental set-up. In sec. 7 we

will discuss our results. Finally, in sec. 8 we will

conclude our work and explain our projects as future

research.

2 LEAST SQUARE FITTING OF

ELLIPSES

2.1 The State of the Art

Two main approaches can be considered for circle de-

tection. The ﬁrst one is to use the Hough Transform

(Yuen et al., 1989). Since spatial perspective alters

the perceived objects, there is the need of calibrating

the camera(s). Then, a pattern recognition algorithm,

such as a simple color detection, can be applied and

subsequently the Hough circle transform can be ap-

plied in order to estimate all the ball’s features.

However, this approach can be complex to be im-

plemented, and even elevate resource consumption.

First, it requires the camera calibration. Moreover,

it can be argued that using a Hough Transform, for

instance, by augmenting the image’s resolution the

computational burden increases as well. Finally, the

Hugh transform needs to be set well, in terms of the

accumulator threshold at the center detection stage

parameter.

The second one is to use ellipse speciﬁc pat-

tern recognition algorithms, such as (Maini, 2006),

(Fitzgibbon et al., 1999). By processing a ball think-

ing of it as it were an ellipse, we overcome the distor-

tion problems. Circles in man-made scenes are almost

always distorted when projected onto the camera im-

age plane, therefore generating ellipses.

Some techniques based on Least Square (LS)

came out in recent years (Fitzgibbon et al., 1999),

(Gander et al., 1994). The principal reason is because

of its computational costs. There are two main kinds

of LS techniques: Those based on the minimization

of the algebraic (Algebraic Distance Least Square,

ADLS) and geometric distance (Geometric Distance

Least Square, GDLS)between the data points and the

ideal curve and those based on the minimization of

the geometric distance ADLSs suffer of high curva-

ture bias (Kanatani, 1994) with the the non-invariance

to Euclidean transformation (Zhang, 1997). However,

GDLSs suffer of being dependent of iterative algo-

rithms (Rosin and West, 1995) as do cluster/voting

(CV) techniques, therefore making them not suitable

for real-time applications (Fitzgibbon et al., 1999).

This is a notable drawback, because iterative algo-

rithms do not have a ﬁxed computational time. Nev-

ertheless, algebraic ﬁtting algorithms may guarantee

a direct one-step convergence. We will focus on this

way, starting from the work of Fitgibbon et Al., called

B2AC, which will be described in the next section

(Fitzgibbon et al., 1999).

2.2 The Original Algorithm

A central conic can be expressed by a second order

equation in its implicit form, as follows in the eq. (eq.

1):

F(x,y) = ax

+ bxy+ cy

+ dx+ ey+ f = 0 (1)

This can also be expressed in the vectorial form:

(x) = x· a =0 (2)

where a =[a, b,c, d,e, f]

is the vector of the equation

coefﬁcients, and x =[x

,xy,y

,x,y,1] is the vector of

the points’ coordinates, both relative to the conic sec-

tion.

Assuming F(a, p

) as the algebraic distance from

the point p

= (x

) to the conic expressed by (eq. 2)

the following non-linear minimization problem has to

be solved (DeSouza and Kak, 2002):

min

(

∑

i=1

F(a,p

)) = min

(

∑

i=1

F(a· p

)

) (3)

REAL-TIME ELLIPSE FITTING, 3D SPHERICAL OBJECT LOCALIZATION, AND TRACKING FOR THE ICUB

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249

In (Fitzgibbon et al., 1999) Fitzgibbon et Al. demon-

strated that solving the problem with the following

constraints gives rise to a unique exact solution:



minkD· ak

· C· a = 1

(4)

where

D =













(5)

and C is a 6 × 6 symmetric matrix where C

(1,3)

= 2,

(2,2)

= −1, and C

(3,1)

= 2.

Now, by using the Lagrange mulltiplier λ and dif-

ferentiating we obtain:



Da− 2λCa = 0

· C· a = 1

(6)

or in the form:







Sa = λCa

· C· a = 1

S = D

(7)

Finally, Fitzgibbon and colleagues demonstrated

that ˜a

= µ

is a unique solution of the system equa-

tions in (eq. 4) (Fitzgibbon et al., 1999), where:

(8)

Therefore, the correspondent afﬁne anti-

transformation (Maini, 2005) needs to be performed

after having found the optimal solution ˜a

Some improvements to the original method

(Fitzgibbon et al., 1999) have been made within the

last years. One deserves particular noticing. In

(Maini, 2005) it has been proposed to compute the

following afﬁne transformation to the input points

before applying the Fitzgibbon’s et Al. algorihm

(Fitzgibbon et al., 1999) :

˜x =

x− x

− 1 ˜y =

y− y

− 1 (9)

where x

= min

i=1

and y

= min

i=1

and:

max

i=1

− min

i=1

max

i=1

− min

i=1

(10)

2.3 Drawbacks and Our Improvement

In 2006 Maini criticized the ill-conditioning of the

scatter matrix S = D

D (eq. 7) and proposed an afﬁne

transformation for solving it by recentering the el-

lipse points within a square with side length equals

to 2 (Maini, 2006). Moreover, in (Maini, 2006) it has

been reported that the algorithm in (Fitzgibbon et al.,

1999) has a speciﬁc source of errors not mentioned in

the paper, and that this causes numerical instabilities,

giving rise to the fact that the closer the data points are

to the ellipse (i.e. the less noise is present), the more

difﬁcult is to locate a unique solution. This results

in the impossibility of having a solution (i.e. a pre-

cise and unique ellipse curve equation) when the data

points lie exactly on, or too close to, the ideal ellipse

curve. In (Maini, 2005), and (Maini, 2006), a resam-

pling procedure has been proposed, that perturbs the

data points with gaussian noise in the case of they are

too close to the ellipse. However, this requires an ex-

cessive computational burden. In fact, he claims that

the procedure must be applied an adequate number

of times M in order to make the algorithm effectively

robust. This makes this approach it not suitable for

real-time applications.

In this work we propose the application of a new

pattern recognition algorithm for the least square of

ellipses we presented in (Greggio et al., 2010) that

overcomes the numerical instability of the original

formulation (Fitzgibbon et al., 1999). Our solution

takes advantage of the improvements given in (Maini,

2006) in terms of the input afﬁne transformation (eq.

9, 10), while using a less expensive computational

perturbing function.

3 LCSE: LEAST CONSTRAINED

SQUARE-FITTING

OF ELLIPSES

3.1 Instability of the Exact Ellipse

Solution

In this section we propose a technique that overcomes

the previous problems. We decided to perturb the

ellipse’s polar transformation by adding a periodic

symmetric function. We apply the data perturbation

only if the case of not stable numerical solution, as

in (Maini, 2006). However, due to our low compu-

tational burden, it does not affect the total computa-

tion sensibly, and can therefore be used any time is

required. The scheme is illustrated in Fig. 1.

The procedure is then described as follows:

• (a) Application of the afﬁne transformation

(Maini, 2006).

• (b) Transformation from the cartesian coordinates

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Figure 1: The RobotCub’s Head. On the left image the head

without the cover is shown, while in the right image the

cover is shown.

Figure 2: the original ellipse after the recentering procedure

(top-left), that represented in polar coordinates (middle), the

polar transformed ellipse with the sinusoidal perturbation

(bottom), and the resultant perturbed ellipse (top-right).

(x,y) to the polar ones (ρ,θ). The point i results:

+ y

= arctan(

); with : x

≥ 0, y

≥ 0

− arctan(

); with : x

≤ 0, y

≥ 0

= π + arctan(

); with : x

≤ 0, y

≤ 0

3π

− arctan(

); with : x

≥ 0, y

≤ 0

(11)

Our aim is to move the points around their initial

position, but maintaining the ellipse average over the

whole polar representation period (2π) within its po-

lar representation. Any symmetric periodic function

with period taken as integer multiplier of 1 added to

the original scattered data leaves the ellipse polar av-

erage unaltered. Therefore, we choose the sinusoidal

function, being continuous, easy to be implemented,

with zero average and inﬁnitely derivable.

• (c) We choose the amplitude equals to A = 0.001

and the frequency equals to f = 1000Hz. The

point i obeys to:

= ρ

+ A· sin(2π fθ

); (12)

• (d) When the ellipse is remapped in cartesian co-

ordinates, its results equally slightly perturbed in-

side and outside its ideal curve, which is the curve

that best interpolates these data. It results, for the

point i:

ˆx

· cos(θ

);

ˆy

· sin(θ

);

(13)

• (e) Now the ellipse is ready to be ﬁtted by build-

ing the design matrix (eq. 5), and by solving the

eigenvalues problem (eq. 7).

• (f) Finally, the afﬁne denormalization transforma-

tion of the point (a) has to be applied.

Fig. 2 shows the original ellipse after the recenter-

ing procedure (top-left), that represented in polar co-

ordinates (middle), the polar transformed ellipse with

the sinusoidal perturbation (bottom), and the resultant

perturbed ellipse (top-right).

3.2 Computational Burden Analysis

Now we analyze the computational complexity of the

algorithm proposed in (Maini, 2006), and our tech-

nique. Low computational complexity means higher

frame rates in real-time applications, and therefore

faster control loops. This results essential in many

actual applications (Vincze, 2001) (Kwolek, 2004)

(Nelson and Khosla, 1994). Our new approach is able

to eliminate the numerical instability that affects the

original algorithm (Fitzgibbon et al., 1999) as (Maini,

2006) does, but greatly faster. We consider N being

the number of points composing the ellipse scattered

data.

Now we will describe: (a.) the resampling pro-

cedure proposed in (Maini, 2006), (b.) our new ap-

proach, and (c.) the ﬁnal comparison between these

two algorithms.

→ a. Resampling procedure: The complete pro-

cedure has been explained in (Maini, 2005). For each

point a gaussian noise component is added. There-

fore, this operation goes with O(N). Therefore the

sequence of operations relative to eq. 7 - 10, has to be

performed. This process goes with O(6N) + O(42N),

repeated for M times. Thus, the resultant complexity

is O(49MN). Finally, an averaging procedure through

all the ellipse data has to be performed, which makes

the overall process going with O(MN)+ O(49MN) =

O(50MN).

→ b. Add sinusoidal perturbation: This adds the

sinusoidal perturbation to the ellipse data after having

been transformed into polar coordinates (originally,

REAL-TIME ELLIPSE FITTING, 3D SPHERICAL OBJECT LOCALIZATION, AND TRACKING FOR THE ICUB

SIMULATOR

251

they are expressed in cartesian representation). Thus

there are three operations to be performed: the ﬁrst

one is the transformation of all the data points from

cartesian to polar representation. This takes O(2N).

After that, the addition of the sinusoidal perturba-

tion takes O(N) operations. Then, the polar coordi-

nates are remapped into cartesian ones, taking O(2N).

Therefore, the whole operation goes with O(5N).

→ c. Computational comparison. In (Maini,

2006) it has been suggested 50 ≤ M ≤ 200. More-

over, it has been reported that EDFE performed better

performances than B2AC for M ≥ 200. However, this

costs an enormous computational burden. Even if M

were equal to 50, our procedure is 500 times faster

than (Maini, 2006). In fact, by comparing the latter

algorithm versus our procedure, it is possible to see

that our procedure is faster of O(50MN)/O(5N) ⇒

10M = 10· 50 = 500 times.

4 THE ICUB ROBOTIC

PLATFORM

The robot is composed of 53 degrees of freedom

(DOFs). Most of them are directly actuated, such as

the shoulders, others are under-actuated, such as the

hands.

In vision, the robotic head design plays an impor-

tant role. Both eyes can tilt (i.e. to move simultane-

ously up and down), pan (i.e. to move simultaneously

left and right), and verge (i.e. to converge or diverge,

with respect to the vision axes). The pan movement

is driven by a belt system, with the motor behind the

eye ball.

An exhaustive explanation about a kinematic and

a dynamic analysis for the upper body structure can

be found in (Nava et al., 2008).

4.1 The ODE iCub Simulation

On the one side, the simulator information is not ex-

haustive, but it is a good approximation for the soft-

ware debugging before using it on the real robot. On

the other side, our algorithm claims to overcome the

original Fitzgibbon’s approach drawback of failing in

detecting the ellipse when the curve lies on the ideal

curve (i.e. is case of noise absence.) (Maini, 2006).

It is clear that image segmentation in the real robot

will never, or very seldom, produce perfect ellipses

after image segmentation, due to all the imperfection

within the real word (light gradients, light contrasts,

color gradients, not regular object shapes, etc), there-

fore testing this ellipse pattern recognition algorithm

to the real robot will not produce comprehensive re-

sults. Contrariwise, the simulator does not present

these artifacts, or at least it limits them.

Tikhanoff et al. developed a completely open

source simulator for the iCub (Tikhanoff et al., 2008),

based entirely on the ODE (Open Dynamic Engine).

We use this simulator in order to test our algorithms.

Fig. 4 shows a print screen of the simulator.

5 SEC::AI034-CUB: THE ROBOT

CONTROLLING TOOL

5.1 The Vision Module

The vision module receives the images from the two

cameras mounted on the iCub head. In order to detect

the ball, and all its features, we implemented a simple

but efﬁcient image processing algorithm. We identify

the ball by means of a color ﬁlter.

(a) The left camera output. (b) The object recognized within the

left camera.

Figure 3: The input image, as seen by the robot within the

simulator with the egocentric view (a) and the same image

with the superimposition of an ellipse, drawn by using the

characteristic parameters obtained by computing the LCSE

(b).

For the identiﬁcation of the blob corresponding to

the ball, we use a connected components labeling al-

gorithm. We assume the largest blob is the ball, so we

look for the blob with the largest area. Subsequently,

we proceeded by applying our LS technique (Greggio

et al., 2010) to the found blob, in order to detect all

the parameters of the curve that describes the bound-

ary of the blob. In Fig. 3(a) the input to the left camera

is presented, i.e. the experimental scenario, while in

Fig. 3(b) output of the algorithm is presented.

5.2 The Motor Control Module

In addition, we implemented a tracking algorithm in

a closed loop. The information received from the vi-

sion module are then processed sent to the motors

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252

of the iCub’s head by means of a velocity control

scheme. that directly commands the head of the robot,

Then, using the forward robot’s kinematics and the

encoders’ information we are able to reconstruct the

target object’s center of gravity (COG) spatial posi-

tion.

. In Fig. 4 a screenshot is depicted, that shows

the testing scenario.

Figure 4: A screenshot depicting the simulated robot in the

3D surrounding environment. Our program uses the en-

coders information to triangulate the position of the centroid

of the object within the simulated space.

5.3 The Kinematics Module

Then, the Denavit-Hartemberg convention for the ob-

ject’s COG coordinates is analyzed. Fig. 5 illustrates

the schematization of the iCub’s neck-head-eyes sys-

tem. Then, tab. 1 contains the relative body kinemat-

ics lenghts, and tab. 2 shows the parameters of the

D-H symbols notation.

Here, the angles represent:

• θ

∗

: Neck pitch - positive up

• θ

∗

: Neck yaw - positive left

• θ

∗

: Eyes tilt - positive up

• θ

∗

: Eyes version - positive left

Then, d

represents the target’s COG distance

from the eyes’ middle axis point E

MID

(see Fig. 5).

This is evaluated with a simple geometrical relation-

ship, as follows:

tan



− E



tan

−1

)

(14)

The target’s COG coordinates (x

COG

) are

evaluated as follows:

The reference system is centered on the ﬂoor plane, at

the center of the pole that sustains the robot. The x axis

evolves along the front of the robot, the y axis runs along

the left of the robot, and the z axis evolves along its height.

Figure 5: Schematization of the iCub’s kinematics. This is

not all the kinematics, of course. We focussed on the head

and neck’s joints.







COG



















(15)

with:







cosθ

∗

cosθ

∗

cosθ

∗

cosθ

∗

+ −sinθ

∗

cosθ

∗

cosθ

∗

cosθ

∗

−cosθ

∗

sinθ

∗

sinθ

∗

−cosθ

∗

sinθ

∗

cosθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

+ sinθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

cosθ

∗

+ −sinθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

−sinθ

∗

sinθ

∗

sinθ

∗

−sinθ

∗

sinθ

∗

cosθ

∗

0 0

−sinθ

∗

cosθ

∗

cosθ

∗

cosθ

∗

cosθ

∗

cosθ

∗

cosθ

∗

−d

sinθ

∗

cosθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

−sinθ

∗

sinθ

∗

−d

cosθ

∗

cosθ

∗

sinθ

∗

cosθ

∗

sinθ

∗

sinθ

∗

−sinθ

∗

sinθ

∗

cosθ

∗

+ d

cosθ

∗

cosθ

∗

cosθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

−d

sinθ

∗

sinθ

∗

sinθ

∗

sinθ

∗

cosθ

∗

cosθ

∗

+ h

0 1







(16)

6 EXPERIMENTS

We performed three types of experiments:

a. The robot localizes a blue cylinder (obtained as

a section of the ball used in the experiment (b)),

and having a negligible height) in front of it; the

cylinder goes away along the x-axis direction at

each trial;

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253

Table 1: Body kinematics lenghts.

Lenght

Symbol Meaning

[SMU]

0.35 E

Eyes y distance: Related to the y axis, it represents the distance between the eyes along their axis

0.36 E

Eyes x offset: Related to the x axis, it represents the distance between the neck pitch axis and the eyes axis

3.50 E

Eyes z offset: Related to the z axis, it represents the distance between the neck yaw axis and the eyes axes

3.01 H

Head z offset: Related to the z axis, it represents the distance between the origin of the reference system and

the neck pitch joint

b. The robot has to evaluate the ball’s radius while

an occlusion hides the object;

c. The robot has to localize a blue ball in front of it.

Table 2: Body kinematics Denavit-Hartenberg parameters.

Link a

0 π/2 h

0 -π/2 0 θ

∗

π/2 h

∗

0 -π/2 0 θ

∗

0 0 θ

∗

Localization is intended in terms of 3D cartesian

coordinates. At each trial the Hough transform, the

B2AC, and the LCSE algorithms are used in order to

evaluate the ball’s center of mass (COM) within the

2D camera images. Therefore this information is tri-

angulated with the encoders’ values in order to deter-

mine the ball spatial position.

Since there is a prospective error, introduced by

the spatial perspective, the ball is not seen as a 2D cir-

cle by the two camera, hence distorted, bringing about

to an artifact during these scenarios experiments that

is not due to the goodness of the three tested algo-

rithms. In order to reduce this effect we tried to iso-

late the perspective error by performing the experi-

ment (a). For each scenario we performed at least 30

trials. The robot stands up and remains in the same

position.

We tested both the precision in localization and

the percentage of success/failure in detection.

7 RESULTS AND DISCUSSION

In the scenario a and b the error between the real and

the evaluated cylinder’s and ball’s position is deter-

mined, while in the scenario c the error between the

real and evaluated ball’s radius is calculated.

7.1 Error Propagation Evaluation

We evaluated the error propagation for the position

detection as follows. All of the terms are measured in

simulator measure unit (SMU). The err

pixel

is the ab-

solute error relative to the value of one square pixel.

In order to evaluate it we referred to the known ball’s

radius. By knowing it (as a ﬁxed value, i.e. 0.17

SMU) and by evaluating it at each measure we can es-

timate the value of a square pixel in SMU (this is the

image resolution at the object distance) as the ratio

between the known radius and the one estimated with

each of the three algorithms considered (i.e. Hough

transform, B2AC, and LCSE).

The errors of the encoders can be considered neg-

ligible within the simulator. Since there is no docu-

mentation on the encoders’ resolution within the sim-

ulator, we considered the accuracy of their informa-

tion approximated to their last digit, which is the

forth one (therefore negligible). Finally the errors due

robot’s lengths need to be considered. Again, there

is no information about the error the lengths of the

robot’s parts have been expressed with. Therefore, in

order to ﬁx their accuracy we analyzed the simula-

tor’s source code. So far, we found that the lengths of

the robot’s parts were expressed with the second digit

of approximation. Hence, we approximated them as

0.01 SMU.

7.2 Scenarios’ Evaluation

As a ﬁrst result, Fig. 6(a) shows the results of the sce-

nario a. With exception for the quadratic error within

the range [2.15 − 2.35], the Hough Transform (HT)

gives rise to the highest error. Both the ellipse ﬁtting

algorithms cause lower error. Speciﬁcally, the B2AC

algorithm is the most precise in terms of quadratic er-

ror, within the ranges [1.2− 1.9], and [2.7− 3.4]. The

LCSE seems to be not the lowest error prone, but it

has a very regular characteristic as a function of the

distance.

The experiment of the scenario b shows a great

linearity between the occlusion of the ball and the er-

ror on its radius evaluation. Fig. 6(b) illustrates the

results of this experiment. Here, the HT gets bet-

ter results within the range [5 % - 20 %] of occlu-

sion. The characteristic is quite linear for all the tech-

niques adopted, with the exception of the cited range,

in terms of a slight decrease from the linear ideal line

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254

(a) (b) (c)

Figure 6: Cylinder’s position error as function of the distance while considering the perspective effect negligible (a), ball

Percentage Error on radius, in % of the radius value (b), and percentage square error, measured in % of the simulator measure

unit (c).

for the HT and a slight increment for both ellipse de-

tection approaches. The two ellipse recognition tech-

niques are more sensitive than the HT to the spatial

perspective.

Finally, the scenario c is discussed. The ellipse

detection approaches give rise to a bigger spatial per-

spective error than the HT within the whole tracking

system, even though the thy also bring about to a big-

ger spatial perspective error than the HT. In 6(c) this

is showed. We did not ﬁlter the results, in order to

keep them as natural as possible. Therefore, we in-

serted three trend lines (one for each technique, each

of them with exponential characteristic) in order to

evidence the most fruiting approach. The B2AC’s and

the LCSE’s trend lines appear superimposed, so that

it is not possible distinguishing them from each other.

The HT’s trend line is the most error prone for balls’

spatial position detection in image processing.

7.3 Some Aspects of the Hough

Transform

It is worth mentioning that the HT depends on some

parameters in order to work correctly. Particularly, we

refer to the accumulator threshold at the center de-

tection stage (ATCDS). The smaller its value is, the

more false circles may be detected, but the higher it is,

the less circles may be detected. The ﬁrst case means

that other curvatures, e.g. artifacts on the ball’s bor-

der caused for instance by inaccuracies of the color

ﬁltering, may be detected as additional objects. How-

ever, the second case can cause the opposite problem.

In our experiments there was only one circular object

within the image, the ball, but setting ATCDS too high

resulted in not detecting it. Therefore, we looked for

the biggest value able to perform all the experiments,

ﬁnding ATCDS = 2. In tab. 3 there are some ATCDS

values: Each one represents the maximum value able

to perform the experiments at some ﬁxed occlusion

and distance values.

However, both B2AC and LCSE algorithms do

Table 3: Maximum value for the accumulator threshold for

getting stability in our experiments.

ATCDS 3 2.6 2.3 2.1 2 2

Occlusion [%] 5 10 15 20 25 30

ATCDS 3.4 3 2.6 2.4 2.1 2

Distance [SMU] 1.2 1.6 2 2.4 2.8 3.2

not present a similar drawback, permitting them to

be used in any situation without any previous setting.

This can be considered a great advantage, since they

do not require any a priori information of the scene to

be analyzed. This is twofold, because allows not only

to build a robust and scene− independent technique,

but also it ﬁts with the concept of cognitive robotics

perfectly.

8 CONCLUSIONS

In this work we presented the ﬁrst implementation of

the LCSE ellipse square ﬁtting algorithm, and we ap-

plied it to a humanoid robotics platform. Moreover,

we implemented a real-time tracking algorithm to lo-

calize an object with the Robot’s stereo vision, and

subsequently we used it to determine the 3D position

of the object’s centroid in the environment. We com-

pared the Hough Transform, the B2AC, and the LCSE

performances in terms of localization precision and

failure in detection in presence of induced artifacts

(such as the ball occlusion by another object) and as

function of the distance of the target. We found that

the B2AC and LCSE give rise to overall more pre-

cise results than the Hough Transform. In the near fu-

ture we plan to apply our techniques to the real iCub

robotics platform, in order to compare and validate

our results with the real robot.

ACKNOWLEDGEMENTS

We thank Dr. Andrea Cini for his contribution in writ-

REAL-TIME ELLIPSE FITTING, 3D SPHERICAL OBJECT LOCALIZATION, AND TRACKING FOR THE ICUB

SIMULATOR

255

ing the DH matrix for the robot forward kinematics.

This work was supported by the European Com-

mission, Project IST-004370 RobotCub and FP7-

231640 Handle, and by the Portuguese Government -

Fundac¸˜ao para a Ciˆencia e Tecnologia (ISR/IST

pluriannual funding) through the PIDDAC program

funds and through project BIO-LOOK, PTDC / EEA-

ACR / 71032 / 2006.

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