Multiple Ellipse Detection by using RANSAC and DBSCAN Method

Kristian Sabo

and Rudolf Scitovski

Department of Mathematics, Josip Juraj Strossmayer University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia

Keywords:

Multiple Ellipse Detection, Clustering, RANSAC, DBSCAN.

Abstract:

In this paper we consider one and multiple ellipse (represented as a Mahalanobis circle) detection problem

on the basis of data points coming from one or several ellipses not known in advance. For solving one

ellipse detection problem two methods are mentioned. These methods are used by solving the multiple ellipse

detection problem. The method proposed in this paper is based on the well-known RANSAC method using the

parameters MinPts and ε from the DBSCAN method. In this way the efﬁciency of choosing the best ellipse

among N ellipses given by the RANSAC method is improved. The local density

ρ =

π|

is determined for each

obtained ellipse

E with circumference |

E| and corresponding cluster

π with |

π| elements . If local density

ρ is

smaller than lower bound

MinPts

2ε

of the local density of the whole set A, the ellipse

E will be dropped. In order

to obtain the ﬁnal solution, an Adaptive Mahalanobis k-means algorithm is applied on the remaining ellipses.

The method is illustrated on several examples with artiﬁcial data point sets and also on a few real images.

1 INTRODUCTION

Ellipse detection problem plays a speciﬁcally sig-

niﬁcant role in different applications such as pat-

tern recognition and computer vision (Akinlar and

Topal, 2013), agriculture, astronomical and geologi-

cal shape segmentation, images analysis in medicine

(Grbi

c et al., 2016), robotics and object detection,

and other image processing (Moshtaghi et al., 2011;

Prasad et al., 2013), etc.

In our paper we consider one and multiple ellipse

detection problem on the basis of a data point set com-

ing from a number of ellipses in the plane, whereby

the edges of these ellipses do not have to be exact or

clear and the number of ellipses does not have to be

known in advance.

The paper is organized as follows. In the next

section the statement of the problem and DBSCAN-

parameters are deﬁned. In Section 3 one ellipse de-

tection problem is considered. 2 Section 4 considers

the multiple ellipse detection problem. The method

based on the RANSAC method with parameters from

the DBSCAN-algorithm (Ester et al., 1996) is proposed.

Two illustrative examples are given. Finally, some

conclusions are given in Section 5.

https://orcid.org/0000-0002-1787-3161

https://orcid.org/0000-0002-7386-5991

2 STATEMENT OF THE

PROBLEM

Given is the set of data points in the plane A = {a

, y

)

: i = 1, . . . , m} ⊂ ∆, ∆ = [a, b] ×[c,d] ⊂ R

a < b, c < d which is considered to be scattered along

multiple ellipses not known in advance and which

should be reconstructed or detected. Additionally, we

suppose that the subset of data points π(E) ⊂ A com-

ing from some ellipse E satisﬁes the “homogeneity

property”, i.e. we assume that the set π(E) is uni-

formly scattered around ellipse E, and the number

ρ(π) =

|π(E)|

|E|

, (1)

where |E| is the length of the ellipse E, will be called

the local density of data point set π(E).

By using the parameters MinPts and ε(A) from

the DBSCAN method (Ester et al., 1996) it is possi-

ble to estimate the lower bound for the local den-

sity of the whole set A. According to (Scitovski and

Sabo, 2019b) we determine the parameter MinPts =

blog|A|c and parameter ε(A) in the following way.

For each a ∈ A we determine radius ε

> 0 of the

smallest disc centered at a and containing at least

MinPts elements of the set A . Then 99.5% quantile

of the set {ε

: a ∈A} is marked with ε(A).

Remark 1. Since for almost all points a ∈A the cor-

responding disc with the center a and the radius ε(A)

Sabo, K. and Scitovski, R.

Multiple Ellipse Detection by using RANSAC and DBSCAN Method.

DOI: 10.5220/0008879301290135

In Proceedings of the 9th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2020), pages 129-135

ISBN: 978-989-758-397-1; ISSN: 2184-4313

129

contains at least MinPts elements of the set A, the

lower bound of the local density ρ(A) of the whole

set A can be estimated with

MinPts

2ε(A)

Now, let us deﬁne the Multiple Ellipse Detection

problem (Akinlar and Topal, 2013; Grbi

c et al., 2016;

Maro

sevi

c and Scitovski, 2015). If the number k ≥ 1

of ellipses is known in advance, the problem is con-

sidered as follows. For the given set of data points

A one should estimate the unknown parameters of el-

lipses E

, q

, ξ

, η

, ϑ

), j = 1, . . . , k:



x(t)

y(t)







+U(ϑ)



cost

sint



, t ∈ [0,2π], (2)

where S

= (p

, q

)

are the centers, ξ

, η

> 0 are

the lengths of semiaxes, ϑ

are the angles and U (ϑ) =



cosϑ −sin ϑ

sinϑ cosϑ



are the matrices of rotation, by solv-

ing the following global optimization problem

argmin

p,q,ξ,η,ϑ

F(p, q, ξ, η, ϑ), (3)

F(p, q, ξ, η, ϑ) =

∑

i=1

min

1≤j≤k

D(a

, E

, q

, ξ

, η

, ϑ

)),

where D is some distance-like function deﬁning the

distance from a point a ∈ A to the ellipse E

(for ex-

ample, like (11)).

If E is understood as the cluster center (E-cluster-

center), then multiple ellipse detection problem (3)

can be interpreted as the problem of searching for a

k-globally optimal partition Π = {π

, . . . , π

}

argmin

Π∈Part(A;k)

F (Π), (4)

F (Π) =

∑

j=1

∑

∈π

D(a

, E

where E

is perceived as the E-cluster-center of clus-

ter π

and Part(A; k) is the set of all k-partitions of the

set A (Morales-Esteban et al., 2014; Scitovski and Sc-

itovski, 2013).

The minimizing function F is nonconvex and

nondifferentiable, but, similarly as in (Scitovski and

Sabo, 2019a), it can prove to be Lipschitz-continuous.

In order to solve the global optimization problem

(3) we can apply one of the known global optimiza-

tion methods (Horst and Tuy, 1996; Paulavi

cius and

Zilinskas, 2014). For example, one could try to

solve the problem by using the well-known DIRECT

global optimization algorithm (Grbi

c et al., 2013;

Jones et al., 1993). However, due to the property of

the DIRECT algorithm to search for all points of the

global minimum, using this algorithm would prove to

be a very inefﬁcient procedure (Scitovski and Sabo,

2019a).

In the case of an unknown number of ellipses us-

ing, for example, incremental method (see Grbi

c et al.

(2016); Bagirov et al. (2011)), an optimal k-partition

for k = 1, 2, . . . is searched for. After that, by using

some indexes adapted for E-cluster-centers (see e.g.

Grbi

c et al. (2016); Scitovski and Sabo (2020)), a par-

tition with the most appropriate number of clusters

can be determined.

There are several methods known for solving this

problem in the literature. Most of them are based

on the Hough transform (Mukhopadhyay and Chaud-

huri, 2015), center based clustering (Maro

sevi

c and

Scitovski, 2015; Moshtaghi et al., 2011; Morales-

Esteban et al., 2014) or geometric method (Isack and

Boykov, 2012; Prasad et al., 2013). The method

EDCircles proposed in Akinlar and Topal (2013) can

be used in real-time applications.

3 DETECTION OF ONE ELLIPSE

We ﬁrst consider one ellipse detection problem in the

plane. An ellipse in the plane can be deﬁned as a set of

points {(x, y)

∈R

: P(x, y) = 0}, where (Fitzgibbon

et al., 1999)

P(x, y) = ax

+ 2bxy + cy

+ dx + ey + f , (5)

where ac −b

= 1. For a given set A the optimal

values of parameters a, b, c, d, e, f , g can be obtained

by solving a constrained linear Least Squares problem

argmin

a,b,c,d,e, f ∈R

ac−b

∑

i=1

(P(x

, y

))

, (6)

and can be written in the standard form (2), as shown

in the next subsection.

3.1 An Ellipse as a Mahalanobis Circle

An ellipse E(S, ξ, η, ϑ), S = (p, q)

, written in the

standard form (2), can be interpreted as a “unit” Ma-

halanobis circle (M-circle)

E = {u ∈ R

: (S −u)

−1

(S −u) = 1)}, (7)

where Σ = (σ

i j

) ∈ R

2×2

is the positive deﬁnite ma-

trix with eigenvalues ξ

, η

. Due to the insurance

of the monotonicity property of the k-means algo-

rithm, a normalized Mahalanobis distance-like func-

tion d

: R

×R

→ R

(u, v;Σ) :=

√

detΣ(u −v)

−1

(u −v)

= ku −vk

(8)

is introduced.

ICPRAM 2020 - 9th International Conference on Pattern Recognition Applications and Methods

130

An ellipse E(S,ξ, η, ϑ) can be written as M-circle

(Scitovski and Sabo, 2019a)

E(S,r, Σ) = {u ∈R

: d

(S, u; Σ) = r

}, (9)

where r

√

detΣ = ξη and conversely, an M-circle

corresponds to the ellipse E(S,ξ, η, ϑ), where

diag(ξ

, η

) = U



√

detΣ



, (10)

U =



cosϑ −sin ϑ

sinϑ cosϑ



, ϑ =

arctan

2σ

−σ

Also, we can now deﬁne the algebraic distance

from the point a ∈ R

to the ellipse E (Morales-

Esteban et al., 2014; Maro

sevi

c and Scitovski, 2015)

D(a, E) = (kS −ak

−r

)

, (11)

where kS −ak

= d

(a, S; Σ). Similarly, we could de-

ﬁne Total Least Squares distance and Least Absolute

Deviations distance (Grbi

c et al., 2016).

3.2 An Ellipse Detection by using

Locally Optimization Method

For recognizing an ellipse on the basis of given set

of data points A which comes from an ellipse not

known in advance we apply some local optimization

method (Newton, Quasi-Newton (Dennis and Schn-

abel, 1996)). For that purpose, it is necessary to

have a good initial approximation. A very good ini-

tial approximation is (see Scitovski and Sabo (2019b,

2020)):

= Mean[A ], (12)

where Σ

∑

a∈A

−a)(S

−a)

and

∑

a∈A

−ak

because

∑

a∈A



−ak

−r



≥

∑

a∈A



−ak

−

∑

a∈A

−ak



Note that the matrix Σ

is a covariance matrix written

using the Kronecker product.

Example 1. Data point set which comes from the el-

lipse E((5, 5), 5, 2,

) is shown in Fig. 1a (Grbi

c et al.,

2016). The corresponding initial approximation ob-

tained by (12) gives the ellipse shown in Fig. 1b. The

ﬁnal solution obtained by Newton (Dennis and Schn-

abel, 1996) method is shown in Fig. 1c.

2 4 6 8 10

(a) Data points

2 4 6 8 10

(b) Initial approximation

2 4 6 8 10

Figure 1: One ellipse detection using local optimization

method.

3.3 Ellipse Detection using the RANSAC

and the DBSCAN Method

RANSAC-method is proposed in the paper Fischler and

Bolles (1981). There are different areas of applica-

tions with this method (see e.g. Cupec et al. (2009);

Isack and Boykov (2012)). In our paper we also apply

this method to solve one and multiple ellipse detection

problem.

An ellipse in the form (5) can be determined on the

basis of several data points from the set A by solving

constrained linear Least Squares problem (6). For ex-

ample, if ﬁve chosen points (x

, y

)

, . . . , (x

, y

)

∈

A do not lie on a line, i.e. if the rank of the matrix







1 x







is equal to 3, we obtain an ellipse candi-

date E, which shall be written in the form of M-circle

E(S,r, Σ). Additionally, if E ⊂ ∆, we assume to have

gotten an acceptable candidate for the ellipse. Finally,

in the ε(A)-neighborhood of the acceptable ellipse we

determine the number of points from the set A. We

repeat the procedure N times (say, 10) and keep that

ellipse

E for which the corresponding set of points is

the largest.

Ellipse

E, written in the form of M-circle

S, ˆr,

Σ),

is a good initial approximation for the ellipse which

will be searched for by solving the local optimization

problem for the function

F(S, r, Σ) =

∑

i=1

D(a

, E(S,r, Σ)), (13)

with the initial approximation {

S, ˆr,

Σ}.

Example 2. The data point set which comes from the

ellipse E((5, 5), 5, 2,

) is shown in Fig. 1a (see also

Multiple Ellipse Detection by using RANSAC and DBSCAN Method

131

Grbi

c et al. (2016)). After N = 8 iterations we got ﬁve

points which resulted in the ellipse shown in Fig. 2a.

The ﬁnal solution obtained by the Newton method is

shown in Fig. 2b and coincides with the ellipse ob-

tained in Example 1.

0 2 4 6 8 10

(a) Initial approximation

2 4 6 8 10

(b) Solution

Figure 2: One ellipse detection using the RANSAC method.

4 THE MULTIPLE ELLIPSE

DETECTION PROBLEM

Now, let us suppose that the data point set A comes

from several ellipses in the plane not known in ad-

vance. We will search for the solution of the multiple

ellipse detection problem by solving global optimiza-

tion problem (3), where D is the algebraic distance

given by (11). Unfortunately, direct application of

the well-known clustering method (see e.g. Domeni-

coni et al. (2016); Scitovski and Scitovski (2013)) is

impossible. Also, the method described in Subsec-

tion 3.2 cannot be applied in this case. For solving

this problem we show a generalization of the method

described in Subsection 3.3. The corresponding algo-

rithm will be called RANSAC for multiple ellipse detec-

tion problem (or abbreviated, RM-algorithm) and will

be described in the following way.

First, we randomly choose 5 points from the set A

not lying on the line. The ellipse determined on the

basis of these points by (6) and contained in rectangle

∆ is an acceptable candidate for the searched ellipse.

We write this ellipse in the form of M-circle E(S, r, Σ).

By repeating the procedure, we assume that we have

found N candidates. In this case the choice of ellipse

in whose ε(A)-neighborhood is the largest number of

points from the set A is no longer an acceptable cri-

terion for the best ellipse as it was in the case of one

ellipse detection (Subsection 3.3).

In multiple ellipse-case we will suppose that the

best ellipse

E has the largest local density (1) of points

in its ε(A)-neighborhood. The cluster

π := {a ∈

A : D(a,

E) < ε(A)} ⊂ A of points from this ε(A)-

neighborhood should be dropped from the set A and

the procedure should be repeated on the rest of the set

A \

π.

We repeat the whole procedure until the number of

the remaining sets becomes smaller than some num-

ber given in advance (for example, 5 MinPts). In that

way we obtain κ ellipses

, j = 1, . . . , κ.

Furthermore, we determine the local density

(

) =

for each pair (

), where |

| is the

number of points in the cluster

, and |

| is the

length (circumference) of the ellipse

which can be

estimated using the well-known Ramanujan approxi-

mation

| ≈π(

)



1 +

10+

√

4−3h



, (14)

where h =

(

−

)

(

)

Note that, for that purpose, it will

be necessary to write the ellipse

in the standard

form

(

) (see Subsection 3.1).

Using the lower bound for the local density of the

set A (see Section 2), the ellipses, for which (see Re-

mark 1)

(

) <

MinPts

2ε(A)

, (15)

will be dropped. We apply the Adaptive Mahalanobis

k-means algorithm to all the remaining ellipses (Grbi

et al., 2016; Maro

sevi

c and Scitovski, 2015; Morales-

Esteban et al., 2014). The algorithm can be described

in following two steps which are repeated iteratively:

Step A: (Assignment step) For each set of mutu-

ally different M-circles E

, r

, Σ

), . . . ,

, r

, Σ

), the set A should be divided

into k disjoint nonempty clusters π

, . . . , π

by using the minimal distance principle;

Step B: (Update step) Given a partition

Π{π

, . . . , π

} of the set A , one can de-

ﬁne the corresponding M-circle-centers

(

, ˆr

) j = 1, . . . , k by using the

methods described in Subsection 3.2 or

Subsection 3.3;

Set E

, r

, Σ

) =

(

, ˆr

) for

j = 1,. . . , k;

Remark 2. Note that, in this way, the remaining el-

lipses will be similar to the original ones. Therefore,

it will not be necessary to use indexes for detecting

the most appropriate partition with ellipses-cluster-

centers in the RM-algorithm.

Note also that searching for new ﬁve randomly

chosen points in the RM-algorithm can be repeated so

long until the local density for obtained ellipse be-

comes greater than the lower bound

MinPts

2ε(A)

Example 3. Let us consider the data point set A

shown in Fig. 3a which comes from three ellipses.

The number of points is |A| = 563, and DBSCAN-

parameters are MinPts = 6, and ε(A) = 0.378. The

lower bound for the local density of whole set A in

this case is 15.9.

ICPRAM 2020 - 9th International Conference on Pattern Recognition Applications and Methods

132

In order to detect ellipses from which the data

point set A comes, we apply the RM-algorithm for el-

lipses detection. First, the red ellipse shown in Fig. 3a

is obtained in 4

attempt. By dropping the points

in its ε(A)-neighborhood, the set of points shown in

Fig. 3b remains. After that, the next red ellipse shown

in Fig. 3b is obtained in 6

attempt. By dropping

the points in its ε(A )-neighborhood, the set of points

shown in Fig. 3c remains. Finally, the third red ellipse

shown in Fig. 3c is obtained in 5

attempt. By drop-

ping the points in its ε(A )-neighborhood, not a single

point of the set A remains. By applying Adaptive Ma-

halanobis k-means algorithm, three ellipses shown in

Fig. 3d are detected.

2 4 6 8 10

(a) Att.4: ρ(π

) = 18.7

2 4 6 8 10

(b) Att.6: ρ(π

) = 19.0

2 4 6 8 10

) = 17.0

2 4 6 8 10

(d) k-means algorithm

Figure 3: Three ellipses detection from Example 3.

Example 4. Let us consider data point set A shown

in Fig. 4a which comes from four ellipses. The num-

ber of points is |A |= 669 and DBSCAN-parameters are

MinPts = 6 and ε(A) = 0.284. Let us show how the

RM-algorithm can be applied in this case. The lower

bound for the local density in that case is 10.6.

In order to detect the ellipses, from which the data

point set A comes, we apply the RM-algorithm for el-

lipses detection. First, the red ellipse shown in Fig. 4a

is obtained in 61

attempt. By dropping the points

in its ε(A)-neighborhood, the set of points shown in

Fig. 4b remains. After that, the next red ellipse shown

in Fig. 4b is obtained in 96

attempt. By dropping

the points in its ε(A )-neighborhood, the set of points

shown in Fig. 4c remains. Furthermore, the next red

ellipse shown in Fig 4d is obtained in 103

attempt.

By dropping the points in its ε(A )-neighborhood, the

set of points shown in Fig. 4e remains. After that,

the next red ellipse shown in Fig 4e is obtained in

attempt. By dropping the points in its ε(A)-

2 4 6 8 10

(a) Att.61: ρ(π

) = 22.6

2 4 6 8 10

(b) Att.96: ρ(π

) = 17.6

2 4 6 8 10

) = 21.0

2 4 6 8 10

(d) Att.167: ρ(π

) = 17.1

2 4 6 8 10

(e) Att.17: ρ(π

) = 9.6

2 4 6 8 10

(f) Att.5: ρ(π

) = 2, 4

2 4 6 8 10

(g) k-means algorithm

Figure 4: Based on the data point set from Example 4, the

RM-algorithm has detected four ellipses.

neighborhood, the set of points shown in Fig. 4f re-

mains. Finally, the next red ellipse shown in Fig 4f

is obtained in 5

attempt. By dropping the points in

its ε(A)-neighborhood, 5 points of the set A remain

remaining and theRM-algorithm is ﬁnished.

The clusters corresponding to the ﬁrst four ellipses

have a local density greater than lower bound 10.6,

and the last two ellipses have a local density less than

the lower bound. Therefore, the last two ellipses will

be dropped, and the k-means algorithm will be ap-

plied to the remaining four ellipses. The results are

shown in Fig. 4g.

Note that the RM-algorithm does not require the

use of indexes for recognizing the most appropriate

partition with ellipse-cluster-centers. One can try to

Multiple Ellipse Detection by using RANSAC and DBSCAN Method

133

improve the efﬁciency of the method by multistarting

the method.

4.1 Real-world Problems

The proposed method for solving the multiple ellipse

detection problem can also be applied to real images.

For that purpose, we carried out the preprocessing of

the edge detection by using the Canny ﬁlter ﬁrst (see

Bradski (2000); Wolfram Research (2016)).

In Fig 5a a fetal head detection on an ultrasound

image is shown. Corresponding edge curves obtained

by Canny ﬁlter in Fig 5b are shown. The red ellipse

in Fig 5a denotes the fetal head. Similarly, in Fig 6,

a detection problem for several cups on the table is

considered.

(a) Detection (b) Canny ﬁlter

Figure 5: Real image.

(a) Detection (b) Canny ﬁlter

Figure 6: Real image.

5 CONCLUSIONS

Solving multiple ellipse detection problem is impor-

tant in many applications. In our paper one and mul-

tiple ellipse detection problem are considered on the

basis of a data point set coming from a number of el-

lipses with noisy edges in the plane. Thereby, we sup-

pose that the subset of data points coming from some

ellipse satisﬁes the “homogeneity property”. For that

situation, a method based on the RANSAC-method is

proposed, whereby the DBSCAN-parameters MinPts

and ε play a signiﬁcantly important role.

It is important to note that the RM-algorithm

does not require the use of indexes for recognizing

the most appropriate partition with ellipse-cluster-

centers. This is the basic advantage of this method

regarding the method EDCircles given in (Akinlar

and Topal, 2013) and method given in (Grbi

c et al.,

2016). Unlike our method, EDCircles does not rec-

ognize an ellipse with semi-axes (ξ, η),

≥ 4 and

cannot detect a single ellipse with a clear edge if

its shape departs signiﬁcantly from a circular shape.

However, our method requires more computing time

than EDCircles.

The method proposed in our paper could be ap-

plied to the case of other geometrical objects too, but

its application is also possible in 3D.

ACKNOWLEDGEMENTS

The author would like to thank the referees and the

journal editors for their careful reading of the pa-

per and insightful comments that helped us improve

the paper. Especially, the author would like to thank

Mrs. Katarina Mor

zan for signiﬁcantly improving the

use of English in the paper. This work was supported

by the Croatian Science Foundation through research

grants IP-2016-06-6545 and IP-2016-06-8350.

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