Arc of ellipse detection for video image registration
Arnaud Le Troter, R´ emy Bulot, Jean-Marc Boi, Jean Sequeira
Laboratoire des Sciences de l’Information et des Syst` emes (LSIS - UMR 6168)
Equipe LXAO (ESIL) - case 925 - 163, avenue de Luminy
13288 Marseille cedex 9, France
Abstract—In this paper, we give a detailed presentation of a
robust algorithm for detecting arcs of ellipse in a binary image.
The characterization of such arcs of ellipse enables the identifica-
tion between some video image elements and the corresponding
landmarks in a 3D model of the scene to be represented. This
algorithm is based on a classical ellipse property that enables
its parameters separation. It provides interesting results even in
noisy images or when these arcs are small and partially hidden.
The Simulfoot project has been developed in the LXAO
group of the LSIS laboratory with the participation of research
scientists in the fields of Computer and Cognitive Sciences.
The goal of this project is to provide a support for team sport
analysis using video sequences, with dedicated applications to
A key step of this process is the 2D-3D registration that
consists in identifying a set of specific elements within the
images and associating them with the corresponding ones in
the 3D model, all these landmarks being referenced in the 2D
model of the field.
This problem can be handled in various ways depending
on video capture conditions. We suppose that we do not have
any knowledge on the capture parameters: the camera has not
been calibrated, it can move and zoom without any constraint,
and we can skip from a camera to another one within a video
sequence. Thus, we will not base the registration process on
the sequence coherence but only on the detection of landmarks
as straight and circular lines that have been drawn on the field
and that appear as straight and elliptic lines in the images.We
take advantage of homography classical properties  and the
representation of these transformations as linear applications
in the homogeneous space to perform the transformation from
the images to the 2D model of the field. A 3 lines and 3
columns matrix represents this transformation.
Thus, our problem is:
• To automatically detect the image area corresponding to
the field (a 2D space in the 3D one).
• To find the matrix that performs the transformation be-
tween the points of the image area and the 2D model.
Research works have been developed in the frame of this
project to automatically detect the Region of Interest (the field)
and the straight lines in this area . When the number of
straight lines in this area is too small, we need to find other
relevant features such as arcs of ellipses that are the projection
of circular lines.
We find points of the primary field itself (i.e. the grass, or
the earth) through the characterization of color coherence in
the region of interest . Thus, by setting to 0 all the points
selected as primary field and all the points that lay out of the
region of interest, we obtain a binary image in which points
to 1 may be associated with the players, the referee, the ball
and the lines drawn on the field . Once straight lines have
been detected in this binary image, we can produce another
binary image in which the points associated with these lines
have been retrieved.
Our goal is to extract arcs of ellipses from this binary image.
II. ARC OF ELLIPSE DETECTION
Many algorithms for detecting ellipses in images have
been developed and published. Most of them use variable
ellipses that automatically match the data (,). We have
developed an algorithm based on the Hough transform applied
to two subsets of separated variables.The most general ellipse
equation is expressed with six parameters and a proportionality
relation on them: that means that ellipses are defined by 5
effective parameters (or by using 5 points). We could think
to use a ”5 to 1” Hough transform to provide such detection
but it would not be acceptable for complexity (the parameter
space would be R5) and non-homogeneity reasons.
Fig. 1.The center O of the ellipse belongs to the line IJ
We have taken advantage of an ellipse geometrical property
to separate this research into two other ones, one in R2and one
in R3), both of them in parameter spaces that have a correct
homogeneity. This property is expressed as follows: ”Let M1
and M2be two points of the ellipse, and let T1and T2be the
tangent lines to the ellipse in these two points. Let I be the
intersection point of T1 and T2, and let J be the middle of
M1M2. Then the IJ line contains the center O of the ellipse”.
This property has already been used for detecting ellipses in
We use it in a different way to detect arcs of ellipse in
noisy binary images. In the following sub-sections, we give
all the details of the proposed algorithm, each sub-section
corresponding to a main step of the algorithm:
• Characterization of relevant couples (point, tangent): we
analyze the neighborhood of each point to decide if this
point supports a straight line; in this case, we consider
that this point belongs to a curve and this straight line is
• Ellipse center detection: we find the ellipse centers by
using the property stated before on the set of selected
• Evaluation of a, b and θ parameters: the ellipse being
centered at the origin, it only depends on three parameters
that are the angle θ between its main axis and Ox, and
the a and b parameters of its normalized expression (a
and b being the half-lengths of its axes)
• Ellipse arcs detection: the ellipse being detected, we find
which points have been associated with its detection and
we analyze their distribution to extract the corresponding
A. Relevant points and tangents characterization
Points with a value equal to 1 in the binary image are
supposed to be related to an ellipse. But they also may be noise
or they may be associated with another scene element. Our first
goal is to estimate, for each point, if it belongs to a curve and,
in this case, to evaluate the tangent to the curve at this point.
The study of point distribution in its neighborhood shows if
there is a main orientation that could be the tangent one. The
method we propose is simple and robust. We first define a
set of masks corresponding to all the possible orientations of
”thick lines” going through its center. The algorithm consists
in applying all these masks to each point (i.e. the mask is
a validation window centered on the studied point) and in
selecting the orientation associated with the mask that provides
the most important number of points, if this number is over
a given threshold S (i.e. below S, we consider the tangent is
A mask is defined by its ”radius” K, its orientation and the
thickness of its line. A mask is a square window of (2K+1) by
(2K +1) points. An orientation is associated with each point
at the edge of the window (in fact, we only consider the points
of the right and the top edges, the two other ones giving the
same orientations). E is the number of parallel lines over and
below the main line when the orientation is given by a point
that belongs to the right edge (and similar for the number of
parallel lines on the right and on the left when the orientation
is given by a point that belongs to the top edge).
The examples below use K = 15 and E = 2.
Fig. 2. Masks for tangent detection
The threshold value S has been chosen equal to K in order
to validate couples (point, tangent) in areas with only a few
points. This algorithm produces a set of N couples (Mi,Ti)
that will be used at the next step to detect the ellipse center.
B. Ellipse center detection
Let us consider now all the couples of couples: ((Mi,Ti),
(Mj,Tj)). We compute the corresponding line Dij= AijBij
where Aij is the middle of segment MiMj and Bij the
intersection point of Tiand Tj.
Because of approximation errors, we do not keep the lines
Dij associated with points Mi and Mj that are too close:
in this case the points Aij and Bij are too close to provide
a reliable line Dij. We could think there may be another
case of error due to the approximation when Ti and Tj are
parallel (Bij being rejected at the infinite): it is not the case
if we compute all the points in the homogenous space (in the
projective space, a point at the infinite has the same status
and is computed in the same way than the others ones). All
these lines focus on the same point if the couples (Mi,Ti)
and (Mj,Tj) belong to a same ellipse. We compute all the
intersections of the Dijand Dkland we study the distribution
of these points in the plane: concentrations areas correspond
to the center of the ellipses to be detected. The method for
finding these concentration areas is classical (Hough space).
C. Ellipse parameterization
Once we have found its center P, we can express the ellipse
equation in a referential of which the origin is P. The most
general expression of this equation is:
Ax2+ Bxy + Cy2= 1
We select those points Mi of the binary image that have
been effectively used to obtain the center P of the ellipse
(points Miused for the construction of Dijwhich contains P
- or which is close to P). Let us call SP this set of points.
Let us consider all the subsets of three points (Mi,Mj,Mk)
from SP. We can compute the coefficient A, B and C of the
corresponding ellipse (that contains Mi,Mjand Mk) and thus:
Mi,Mj,Mk∈ Sp: (Mi,Mj,Mk) → (A,B,C)ijk
We could think to use a ”3 to 1” Hough transform on the
(A,B,C) coefficients but we do not have any control on the
homogeneity of the corresponding space. Thus, we chose to
work in the (a,b,θ) space where θ is angle (Ox,∆) - ∆ being
the ellipse main axis - and where a and b are the half-lengths
of its axes. The expression of (a,b,θ) as a function of (A,B,C)
A + C −?B2+ (A − C)2
A + C +?B2+ (A − C)2
A − C+kΠ
As we did before to find the ellipse center, we use a classical
procedure to find the concentrations (there will be only one
in the frame of the problem we are interested in) and thus the
final coefficients (a,b,θ)P.
D. Arc of ellipse detection
For each ellipse detected in the image, we have its parame-
ters (center, orientation and axes lengths) and the set of points
that are in its neighborhood. In order to characterize the arcs
of ellipse associated with this set of points, we apply to these
points the homography that transforms the ellipse into the
circle centered at the origin and which radius is 1 (unit circle).
This transformation is expressed in the homogeneous space
as a 3 lines and 3 columns matrix. It takes into account
• A move from P (ellipse center) to O (origin)
• A rotation of −θ
• A scaling of 1/a in the Ox direction
• A scaling of 1/b in the Oy direction
After this transformation, all the selected points are close
to the unit circle. First, we eliminate the points that are not
very close to this circle: we only keep the point that are at a
distance to the origin belonging to [1-?,1+?] (? being chosen
as 0.1). Then, we sort all the remaining points by increasing
values of θ. Finally, we segment this set θ values by merging
all of the points (or θ values) of which value difference is less
that δθ (we have chosen δθ = 10obut any other choice should
also be valid as a representation with a different granularity).
The images below illustrate all this stpdf: the ellipse detec-
tion, the point selection and distribution analysis, and finally
the arc detection.
Fig. 3. Ellipse detection, homography, arc detection
In this section, we present some results we obtained in the
frame of the Simulfoot project and how we exploit them to
provide the registration.
Fig. 4.Arc of ellipse detection close to the goal area
Figure 4 shows us the detection of an arc of ellipse.
There are two arcs of ellipse and a full ellipse on a soccer
field. Knowing that we have an arc of ellipse and having its
orientation gives us strong information on which part of the
field is visualized. But there is to take care: if the detected
arc(s) is (are) close to the image boundary it may have been
clipped and then, the rules for characterizing the part of the
field are different.
Fig. 5.Ellipse detection in the center of the field
Figure 5 shows us the detection of a full ellipse that
corresponds to the circle drawn at the center of the field. In
fact, it is not a full ellipse but a set of arcs that all belong to
same ellipse due to the players who hide parts of it.Studying
the logic links of the resulting arcs and having knowledge on
the scene enable us to determine which part of the field is
The detection of arcs of ellipse is a key process for
image to model registration when analyzing team sports video
sequences. This is due to the presence of circles (or arcs of
circle) on the field in most cases (basket-ball, handball,).
We have designed a new algorithm that enables such detec-
tion in a noisy binary image. This algorithm is robust and fast
enough for the Simulfoot project applications.
We will now focus our research works on the automatic
understanding of the scene in relation with the different
elements extracted from the images (ellipses, lines, ) and the
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