A Contrario Elliptical Arc, Circular Arc and Line Segment Detection

Boshra Rajaei

1,2

and Rafael Grompone von Gioi

Sadjad University of Technology, Mashhad, Iran

CMLA, ENS Paris-Saclay, France

Keywords:

Ellipse Detection, Line Segment Detection, A Contrario Theory.

Abstract:

In this paper, we propose a joint elliptical arc, circular arc, and line segment detector based on the a contrario

statistical approach. Our method is an extension of the ELSDc method, recently proposed for line segment

and elliptical arc detection. The main contribution is a more general geometrical model, which allows the

joint evaluation of the best combination of elliptical arcs, circular arcs, and line segments that corresponds to

a given contour. Different interpretations in terms of these elements are tried for the whole contour, instead

of locally as it is done in ELSDc. In addition, several minor improvements were performed to the heuristic

algorithm used to propose candidates. The performance of the proposed method is compared to the original

one on synthetic and real images.

1 INTRODUCTION

The extraction of geometrical primitives correspond-

ing to edges is a compact way of representing the ge-

ometrical contents of a digital image. Several such

algorithms were proposed in the past, among others,

for full lines and line segment detection (Bonci et al.,

2005; Ji et al., 2011; Alpatov et al., 2015), for cir-

cles and circular arc detection (Goulermas and Liat-

sis, 1999; Yao and Yi, 2016), and for ellipses and el-

liptical arc detection (Lu and Tan, 2008; Chia et al.,

2010; Arellano and Dahyot, 2016; Grbi

c et al., 2016).

Naturally, this kind of methods are exposed to both

false positive and false negative, due to the inevitable

noise of real images and the difﬁculty in deﬁning

these primitives and especially their interactions.

In the past decade, Desolneux et al. (Desolneux

et al., 2000; Desolneux et al., 2008) proposed the so-

called a contrario methodology to set automatically

the detection thresholds in order to control the num-

ber of false detections. This statistical approach is

based on multiple testing procedures (Gordon et al.,

2007) and the main idea is that an observed structure

is meaningful only when the relation between its parts

is too regular to be the result of an accidental arrange-

ments of independent parts. This approach led to

several automatic feature detectors which are robust

to image noise. This includes detectors of line seg-

ments (Grompone von Gioi et al., 2010; Akinlar and

Topal, 2011), circles (Akinlar and Topal, 2013), sym-

metric patches (P

atr

aucean et al., 2013), regular ar-

rangement of line segments (Rajaei et al., 2016), sub-

jective contours (Rajaei et al., 2017), or point align-

ments (Lezama et al., 2014), to mention a few. In all

these cases, the a contrario approach was used to set

the detection thresholds for a single kind of geometric

primitive.

Recently, an a contrario method was proposed

for joint detection of line segments and elliptical

arcs (P

atr

aucean et al., 2017; P

atr

aucean et al., 2012).

The number of false detections is controlled by using

a contrario models for both, line segments and ellip-

tical arcs. When these two interpretations both satisfy

the detection threshold on the same part of the image,

the same modeling is also used to compare them and

to select the interpretation which corresponds better to

the observation. Therefore, this model selection step

helps to reduce the error of misinterpreting one primi-

tive by another. More precisely, the ELSDc algorithm

ﬁrst uses some heuristics to propose detection candi-

dates. The aim of these heuristics is to obtain a high

recall without spending too much computational time;

rejecting false detections due to noise is the task of

the a contrario validation step. Three types of events,

polygonal line, elliptical arc and circular arc, are con-

sidered. A separate a contrario validation criterion is

used for each of these events, and a Number of False

Alarms (NFA) is assigned to each one, indicating its

degree of accidentalness in a random model. Finally,

if more than one event is declared meaningful, the one

538

Rajaei, B. and von Gioi, R.

A Contrario Elliptical Arc, Circular Arc and Line Segment Detection.

DOI: 10.5220/0009172005380545

In Proceedings of the 15th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2020) - Volume 4: VISAPP, pages

538-545

ISBN: 978-989-758-402-2; ISSN: 2184-4321

Figure 1: A sequence of rectangles and a candidate arcline

event composed of one line segment (a polygonal line of

one part) and an elliptical arc.

with best NFA value is kept as the detected primitive.

In practice, the ELSDc algorithm does not always

manage to produce the expected result. This is mainly

due to the fact that it starts by cutting a curve into parts

that does not change the curvature sign, i.e., the curve

is cut at inﬂection points. Then, each of those parts is

evaluated as either one polygonal line, one elliptical

arc, or as one circular arc. But a piece of curve with-

out curvature sign changes is not necessarily compose

of just one of those primitives. For example, a line

segment concatenated to an elliptical arc can share the

same curvature sign, see Figure 1. In such cases, the

polygonal approximation would usually get the best

ﬁt and be kept. As a result, ELSDc has a tendency

to prefer the polygonal interpretation over circular or

elliptical arcs.

In this paper we propose an algorithm that follows

the same general scheme of ELSDc but using a richer

geometrical event. A whole curve (possibly includ-

ing curvature sign changes) is interpreted by what we

call an arcline, namely a sequence of line segments,

circular arcs and elliptical arcs. Again, heuristics are

used to propose candidate arcline events, i.e., different

ways of cutting the curve into line segments, circular

arcs and elliptical arcs. As in ELSDc, an a contrario

validation step rejects false detections due to noise,

and the valid interpretation with the best NFA value

is selected. As it is classic in model selection, this

step takes into consideration the complexity of the in-

terpretation and how well it ﬁts the image data.

Wolters et al. (Wolters and Koch, 2017) also pro-

posed a detector for combined extraction of line seg-

ments and elliptical arcs which, by modeling the

topological relationships of the individual features in

a graph, describes more complex geometric struc-

tures over multiple connected primitives. The authors

showed that their algorithm outperforms ELSDc over

edges with multiple structures. However, on simple

edges the experimental results show that ELSDc has

a better performance. Our experiments will compare

the proposed method to the ELSDc (P

atr

aucean et al.,

2017) and to Wolters et al.’s method (Wolters and

Koch, 2017).

This paper is organized as follows. The next sec-

tion outlines the a contrario approach and section 3

summarizes the ELSDc algorithm. The main contri-

bution of this paper is introduced in section 4, where

the a contrario formulation for the arcline event is

presented. Then, the proposed algorithm is described

in section 5 and the experimental results are discussed

in section 6. Finally, section 7 concludes the paper.

2 THE A Contrario APPROACH

The a contrario theory (Desolneux et al., 2000; Des-

olneux et al., 2008) is a statistical framework used to

set detection thresholds automatically in order to con-

trol the number of false detections. It is based on the

non-accidentalness principle (Witkin and Tenenbaum,

1983; Lowe, 1985) which informally states that there

should be no detection in noise. In the words of

D. Lowe, “we need to determine the probability that

each relation in the image could have arisen by acci-

dent, P(a). Naturally, the smaller that this value is,

the more likely the relation is to have a causal inter-

pretation” (Lowe, 1985, p. 39).

A stochastic background model H

needs to be

deﬁned, where the structure of interest is not present

and can only arise as an accidental arrangement. For

example, the ELSDc algorithm, as well as the algo-

rithm proposed here, is based on the orientation of the

image gradient. Thus, the background model H

as-

sumes that the gradient orientations at each pixel are

independent random variables, uniformly distributed

in [−π, π); under such model, a region of the image

where the gradient orientation follows a regular struc-

ture would be a rare accident.

We also need to deﬁne a family of events of in-

terest T . For feature detection the family of events is

the set of all the geometrical events considered, i.e.,

all the line segments, circular arcs, etc., considered

in the image domain. Then, we need to assess the

accidentalness of a candidate feature. For example,

if a line segment is present in an image, the gradient

orientation at the corresponding position would be or-

thogonal to the line segment. Then, given a candidate

line segment, one measures how well the image gradi-

ent corresponds to the candidate event, and we need to

evaluate the probability of observing by chance such

a good agreement. A rough agreement could arise

just by chance and thus does not correspond to an

interesting event; inversely, a good agreement would

be rare and suggest a casual reason instead of just a

lucky accident. In other words, when this probability

is small enough, there exists evidence to reject the null

hypothesis and declare the event meaningful. How-

A Contrario Elliptical Arc, Circular Arc and Line Segment Detection

539

ever, one needs to consider that multiple candidates

are tested. If 100 tests were performed, for example,

it would not be surprising to observe among them one

event that appears with probability 0.01 under random

conditions. The number of tests N

needs to be in-

cluded as a correction term, as it is done in the statis-

tical multiple hypothesis testing framework (Gordon

et al., 2007).

Following the a contrario methodology (Desol-

neux et al., 2000; Desolneux et al., 2008), we deﬁne

the Number of False Alarms (NFA) of an event e ob-

served up to an error k(e) as:

NFA(e) = N

·P

(e) ≤ k(e)

, (1)

where it is evaluated the probability of obtaining in

the background model H

an error K

(e) smaller

or equal to the observed one k(e). The smaller the

NFA value, the more unlikely the event e is to be ob-

served by chance in the background model H

; thus,

the more meaningful. The a contrario approach pre-

scribes to accept as valid detections the candidates

with NFA ≤ ε, for a predeﬁned ε value. It can be

shown (Desolneux et al., 2000; Desolneux et al.,

2008) that under H

, the expected number of tests

with NFA ≤ ε is bounded by ε. As a result, ε corre-

sponds to the mean number of false detections under

. In most practical applications the value ε = 1 is

suitable and we will set it once and for all.

3 THE ELSDc ALGORITHM

The ELSDc algorithm (P

atr

aucean et al., 2017) em-

ploys the a contrario approach to detect more than

one image structure simultaneously. The method con-

siders polygonal line, elliptical arc, and circular arc as

three geometrical primitives. The detection strategy

goes through three main steps: 1) candidate genera-

tion; 2) candidate validation; and 3) model selection.

Candidate Generation. A heuristic procedure is

used to propose candidates instead of testing all possi-

ble ones, which would requires too much time. Start-

ing from one seed pixel, a region growing procedure

recursively merges all neighboring pixels sharing the

same gradient orientation up to a precision; this step

is based on the heuristics used in (Grompone von Gioi

et al., 2010). A rectangle covering the region of pixels

found determines a line segment with a given width.

Then, a second region is started from a new seed pixel

at one end-point of the rectangle. This procedure is

repeated until no new region is found. The result is a

chain of rectangles roughly following an image con-

tour. The chain of rectangles is then cut into parts

Figure 2: ELSDc polygonal line and elliptical arc candi-

dates. A polygonal line is deﬁned as a sequence of rectangle

end-points with a common width. An elliptical arc is de-

termined by eight degrees of freedom: ellipse center, axes,

orientation, delimiting angles and width. (Figure reprinted

from (P

atr

aucean et al., 2017)).

such that consecutive rectangles in one part turn in the

same direction; this leaves parts which have the same

curvature sign. Finally, three candidates are generated

for each chain part: a polygonal line p

composed of

d line segments, determined by the central lines of

the rectangles in the chain; a circular arc c obtained

by ﬁtting the gradient orientations on the pixels cor-

responding to the rectangles; and an elliptical arc t,

similarly obtained by ﬁtting the gradient orientations

to an elliptical arc. Figure 2 shows an example of

a part of a chain of rectangles, corresponding to a

polygonal candidate and a ﬁtted elliptical candidate.

See (P

atr

aucean et al., 2017) for more details.

Candidate Validation. After candidate generation,

the ELSDc algorithm employs an a contrario valida-

tion step to reject non-meaningful candidates accord-

ing to equation 1. Let us call I the input image of size

n ×m. Let S(e) denotes the set of pixels supporting

an event e, that is, the set of pixels in the correspond-

ing positions of the image. Let l(e) be the size of the

support region S(e). The agreement error between a

candidate event e and the image I is measured by the

sum of the normalized angle difference between the

gradient orientation and the normal direction to the

event at each pixel of the support region S(e):

k(e) =

∑

i∈S(e)



AngleDifference



∇I(i), dir

⊥e

(i)





(2)

where the sum is over all pixels i belonging to the

support region S(e), ∇I(i) is the gradient of the im-

age at pixel i, and dir

⊥e

(i) corresponds to the normal

direction to the geometrical event e at position i. The

agreement error k(e) takes values between zero (when

there is a perfect agreement between the event and the

image gradient) and l(e) (for the worst possible agree-

ment) (P

atr

aucean et al., 2017).

Under the background model H

deﬁned in sec-

VISAPP 2020 - 15th International Conference on Computer Vision Theory and Applications

540

tion 2, the normalized angle differences at each pixel

are independent random variables following a uni-

form distribution in [0, 1]. Thus, K

(e) follows

an Irwin-Hall distribution of l(e) parameters, which

leads to the probability term in Eq. 1 given by

(e) ≤ k(e)

l(e)!

bk(e)c

∑

j=0

(−1)



l(e)



(k(e) − j)

l(e)

(3)

where bxc is the largest integer not greater than x

and





is the binomial coefﬁcient. To simplify the

presentation, we will use the upper-bound given by

ﬁrst term, P

(e) ≤ k(e)

≤

[k(e)]

l(e)

l(e)!

, which allows

a more compact formula and is accurate enough for

our needs, see (P

atr

aucean et al., 2017).

We will focus here on the main two events: polyg-

onal lines (p

) and elliptical arcs (t) (circular arcs

are treated as a particular case of elliptical event).

In each case, the agreement error en Eq. 2 is com-

puted relative to the corresponding primitive, polyg-

onal line or elliptical arc. The difference in formu-

lation is in the number of events (N

) considered in

each family. The number of tests are approximated by

d+1

(mn)

(d+1)+

for a polygonal line of d segments,

and 18(mn)

for an elliptical event (recall that the im-

age size is n ×m), see (P

atr

aucean et al., 2017). Then,

the NFA values for polygonal line and elliptical arc

events are obtained from Eq. 1:

NFA(p

) = 2

d+1

(mn)

(d+1)+

[k(p

)]

l(p

)

l(p

, (4)

NFA(t) = 18(mn)

[k(t)]

l(t)

l(t)!

. (5)

As usual, an event with NFA ≤ ε is considered as a

detection, with ε = 1.

Model Selection. For some sequence of rectangles,

it could happen that more than one candidate (p

, c,

or t) obtain NFA ≤ ε and are thus meaningful. As

stated in section 2, the smaller the NFA value, the

more meaningful the event. Thus, the ELSDc algo-

rithm simply selects and keeps, among those three,

the interpretation with the smallest NFA value.

4 THE ARCLINE EVENT

As explained in section 3, the ﬁrst step of the ELSDc

algorithm cuts a chain of rectangles into parts in

which there is no change of curvature sign; then, each

of these parts is evaluated as either a polygonal line, a

circular arc, or an elliptical arc. But a piece of curve

without curvature sign changes can corresponds to a

sequence of line segments and elliptical arcs, see ﬁg-

ure 1 for an example. In such cases, ELSDc would

produce either one elliptical arc with a rough ﬁt or,

more often, a polygonal line.

To overcome this issue, we deﬁne a new event

which we call an arcline that consists in a sequence

of line segments, circular arcs, and elliptical arcs:

Deﬁnition 1. An arcline a

g,h

is an ordered set of f

primitives, where g of them are circular arcs, h are

elliptical arcs and the remaining f −g −h are line

segments, and such that consecutive elements in the

chain share end-points. The geometrical conﬁgura-

tion is completed with a width value, deﬁning a region

of pixels inside each geometrical primitive.

The rest of this section is dedicated to the presen-

tation of the a contrario formulation to validate ar-

cline events. Next section will describe an algorithm

similar to ELSDc but using arclines as geometrical

primitives.

The same background model H

deﬁned in sec-

tion 2 is used and the agreement error k(a

g,h

) between

an arcline a

g,h

and the image is measured as in Eq. 2,

where dir

⊥a

g,h

(i) corresponds to the normal direction

to the primitive of a

g,h

at position i, i.e., either a line

segment, a circular arc, or an elliptical arc. Again, the

only difference in the formulation is in the number of

test N

in Eq. 1.

The number of elements in the family must in-

clude any relevant arcline event. First, we start by

dividing the family according to the number of ele-

ments in the chain, f , and dividing the total number

of allowed false detection ε accordingly. The arclines

with just one primitive would be allowed

false de-

tections in H

; the arclines composed of two primi-

tives would be allowed

false detections in H

; gen-

erally, the arclines composed of f primitives would

be allowed

false detections in H

. Thus, an arcline

composed of f primitives would be declared mean-

ingful when

( f ) ·P

g,h

) ≤ k(a

g,h

)

≤

, (6)

where N

( f ) is the number of arclines composed of

f primitives. As a consequence, the total number of

false detections in H

would be bounded, as desired,

∞

∑

f =1

= ε. (7)

The condition in Eq. 6 is equivalent to

·N

( f ) ·P

g,h

) ≤ k(a

g,h

)

≤ ε, (8)

A Contrario Elliptical Arc, Circular Arc and Line Segment Detection

541

where 2

appears as a factor in the number of tests.

In an arcline composed of f primitives, each of

the f primitives can be either a line segment, a circu-

lar arc, or an elliptical arc. Thus, there are 3

possi-

ble conﬁgurations. Again, we can divide ε among all

these 3

conﬁgurations, what is equivalent, as before,

to including a factor 3

in the number of tests.

Now, for an arcline composed f primitives in one

of the 3

possible conﬁgurations, there are f + 1 end-

points in the chain, and considering pixel precision

in an n ×m image, there are (nm)

f +1

possibilities.

For each of the g circular arcs (whose end-points are

determined), there is still one degree of freedom to

determine its curvature. Assuming roughly

√

mn op-

tions for each one, this adds a factor (nm)

. Similarly,

an elliptical arc still has three degrees of freedom once

the end-points are determined, adding a factor (nm)

for the h elliptical arcs. Finally, we need to determine

the arcline width; again, we will assume that there

are about

√

mn possible values. Then, there are about

(nm)

f +

3+g+3h

arclines to be considered.

All in all, the NFA for the arcline event is given by

NFA(a

g,h

) = 2

(nm)

f +

3+g+3h

k(a

g,h

)

l(a

g,h

)

l(a

g,h

(9)

As usual, an arcline a

g,h

is declared meaningful when

NFA(a

g,h

) ≤ ε. Also, the smaller the NFA(a

g,h

)

value, the more meaningful the arcline. Thus, the

model selection step will keep the arcline with the

smallest NFA value as the best interpretation.

5 THE ALD ALGORITHM

The ArcLine Detector (ALD) shares the same three

step strategy of ELSDc: 1) candidate generation; 2)

candidate validation; and 3) model selection. Here all

the candidates are arclines with different conﬁgura-

tions. Note that the candidate generation step aims at

avoiding false negatives while not spending too much

computational time; missing detections are in part due

to this heuristic step, when failing to propose the right

candidate. However, accepted detections must satisfy

the statistical test imposed by the a contrario theory;

the same is true for the model selection step.

Algorithm 1 outlines the main steps of the ALD

method. Starting from seed pixels, a sequence of rect-

angular regions r is obtained by an iterative region

growing procedure (steps 1 to 4). Then, all different

ways of partitioning the sequence into line segments,

circular arcs, and elliptical arcs are examined (steps 5

to 12). For each conﬁguration a

g,h

, the NFA value is

computed, which evaluates its agreement with the in-

put image I (step 9). Model selection is performed by

keeping the arcline a

with the smallest NFA value

(steps 10 to 12). Finally, the arcline a

is stored if it

is meaningful, i.e., if NFA ≤ ε with ε = 1 (steps 13

to 15). The supporting pixels of a validated arcline

are prevented from being used again to avoid spend-

ing time on data already handled.

Algorithm 1: Arcline detector algorithm.

input : image I

output: a set of validated arclines A

1 I

← G

? I Gaussian blur

2 Seeds ←{s, |∇I

(s)| > ρ} set of seed pixels

3 for s ∈ Seeds do loop on seed pixels

4 r ← RegionGrowing(s,∇I

, θ, γ, δ)

5 P ← GeneratePartitions(r)

6 nfa

← ∞

7 for partition ∈ P do loop on partitions

8 a

g,h

← FitArcline(partition, r, ∇I

, α)

9 nfa ← NFA(a

g,h

) NFA of arcline, Eq. 9

10 if nfa < nfa

then

11 nfa

← nfa keep best NFA

12 a

← a

g,h

keep best arcline

13 if nfa

≤ 1 then meaningful arcline found

14 add a

to A

15 remove supp. pixels of a

from Seeds

Given the richness of the arcline event, generat-

ing relevant candidates is quite challenging; a set of

heuristic methods are used. The main ideas are simi-

lar to the ones in the ELSDc algorithm, but some im-

provements were introduced. As in ELSDc, a Gaus-

sian blurred version I

of the input image is com-

puted (step 1) to be used for candidate generation;

this makes the region growing process more robust

to noise and small irregularities. However, the NFA

value used for validation (step 9) is computed on the

non-blurred input image I. To select good starting

points which probably belongs to edges, the seeds are

selected as pixels with gradient magnitude larger than

a ﬁxed value ρ (step 2).

The region growing procedure (step 4) groups to-

gether pixels that share the same gradient orientation

up to a tolerance θ. Those pixels must, nevertheless,

also satisfy the condition that their gradient magni-

tude be larger than γ. In ELSDc, ρ and γ have the same

value, but here ρ is more strict for selecting good seed

pixels, while allowing a more relaxed condition γ on

VISAPP 2020 - 15th International Conference on Computer Vision Theory and Applications

542

Figure 3: An instance of an edge with two low contrast re-

gions. The region growing algorithm used in ELSDc splits

the support region into three regions. This kind of regions

are better handled by the hysteresis-like approach with two

gradient thresholds proposed here: a week threshold for re-

gion growing to pass the gaps and a stronger one for actual

candidate generation and validation.

the neighbor pixels. Figure 3 shows a simple example

were the new strategy helps to improve the result. A

small gap in edge points, that happens frequently in

real images due to changes in intensity, stops the re-

gion growing; the more permissive threshold γ, there-

fore allows to jump gaps.

As in ELSDc, the rectangles are cut until a min-

imum density of support pixels of δ is reached. But

here, only pixels with gradient magnitude larger than

α times gradient magnitude of the seed pixel s are

considered when ﬁtting the arcline a (step 8) to make

sure that only strong edge points determine the ﬁnal

detection. Besides this improvement, the ﬁtting of el-

liptical and circular arcs to the region r is performed

by the method proposed in (P

atr

aucean et al., 2012).

Several strategies are used to reduce the computa-

tional burden. An upper bound to the number of parts

in the partitions is imposed, currently set to 10. Also,

the validation step is straightforwardly parallelized.

The parameter of the candidate generation heuristics

are set to σ = 1, θ = 22.5

◦

, ρ = 10, γ = 5, δ = 0.3

and α = 0.2. For more details, we refer the reader

to (P

atr

aucean et al., 2017) and to the publicly avail-

able source code of the proposed method (see below).

6 EXPERIMENTS

In this section, we examine the ALD algorithm over

a few synthetic and real images. The interested read-

ers may refer to the web page of the article

and use

the online demo for trying out the algorithm on ar-

bitrary images. For comparison purposes, two recent

http://dev.ipol.im/

∼

jirafa/ald/

algorithms which both detect straight lines and ellipti-

cal arcs simultaneously are used, ELSDc (P

atr

aucean

et al., 2017) and Wolters et al (Wolters and Koch,

2017).

Figure 4 shows the detections of ELSDc and ALD

algorithms on three images with multiple line seg-

ment and elliptical arc primitives along edges. Here,

the ELSDc algorithm is executed using the public

available source code of the authors. On edges con-

taining both, straight and non-straight structures, the

ELSDc algorithm is forced to choose between either

an elliptical arc or a polygonal line. In many cases the

decrease in agreement error in polygonal line event

dominates over the increased number of tests. There-

fore, the ELSDc algorithm chooses polygonal line

and, comparing to ALD, misses many curvy struc-

tures. This effect is visible in, for instance, the inner

contour in parking sign image and in the outer circular

arc in the dolphin image.

Additionally, the double gradient thresholds (ρ

and γ) in the region growing and validation steps

in ALD result in longer regions and give the algo-

rithm the chance of using a more suitable primitive

to model the edge. This happen in the clocks im-

age where ELSDc misses some curves from the outer

clock boundaries and groups the pixels in different re-

gions.

The second experiment compares the algorithms

using the three real images from (Wolters and Koch,

2017)

, see Figure 5. The ALD algorithm produces

a good synthesis between both methods, detecting as

many good line segment structures as ELSDc, and as

many good elliptical or circular arcs as Wolters et al.’s

method.

7 CONCLUSIONS

We proposed an a contrario model for line segment,

circular arc, and elliptical arc detection in digital im-

ages. The arcline geometrical event is used to express

an edge in terms of those primitives and select the

best interpretation for the whole edge. An heuristic

algorithm is also proposed to cope with the computa-

tional burden of ﬁnding good candidates. The results

compares favorably to state-of-the-art methods. Fu-

ture work will concentrate on improving the ellipse

ﬁtting step, which in some cases leads to small local-

ization offsets.

We did not have access to the source code of the

method by Wolters et al.; to have a fair comparison, the re-

sults of that algorithm are reprinted from the original paper.

A Contrario Elliptical Arc, Circular Arc and Line Segment Detection

543

Figure 4: Detected image structures on the synthetic ‘parking sign’ image, and real ‘dolphin’ and ‘clocks’ images. From left

to right are the input images, the result of ELSDc (P

atr

aucean et al., 2017), and the result of the proposed ALD algorithm

(red: elliptical or circular arcs, blue: line segments and, green: cut points).

Figure 5: Detected image structures on the images ‘building’, ‘window’ and ‘roof’. From left to right are the input images,

the results of ELSDc (P

atr

aucean et al., 2017), Wolters et al. (Wolters and Koch, 2017) (reprinted from (Wolters and Koch,

2017)), and the proposed ALD algorithm (red: elliptical or circular arcs, blue: line segments, and green: cut points).

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544

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