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A Probabilistic Measure of Circularity

Ana Marcela Herrera-Navarro1,

Hugo Jim´enez-Hern´andez2,andIv´an Ram´on Terol-Villalobos3

1Facultad en Ingenier´ıa, Universidad Aut´onoma de Quer´etaro,

76000, Quer´etaro, M´exico

2CIDESI, Av. Playa Pie de la Cuesta No. 702, Desarrollo San Pablo, 76130,

Quer´etaro, M´exico

3CIDETEQ, S.C., Parque Tecnol´ogico Quer´etaro S/N,

San Fandila-Pedro Escobedo, 76700, Quer´etaro, M´exico

anaherreranavarro@gmail.com

Abstract. The circle is a useful morphological structure: in many situ-

ations, the importance is focused on the measuring of the similarity of a

perfect circle against the object of interest. Traditionally, the well-known

geometrical structures are employed as useful geometrical descriptors,

but an adequate characterization and recognition are deeply aﬀected by

scenarios and physical limitations (such as resolution and noise acquisi-

tion, among others). Hence, this work proposes a new circularity measure

which oﬀers several advantages: it is not aﬀected by the overlapping, in-

completeness of borders, invariance of the resolution, or accuracy of the

border detection. The propounded approach deals with the problem as

a stochastic non-parametric task; the maximization of the likelihood of

the evidence is used to discover the true border of the data that repre-

sent the circle. In order to validate the eﬀectiveness of our proposal, we

compared it with two recently eﬀective measures: the mean roundness

and the radius ratio.

Keywords: Measure, Circularity, Shape, Disk, Topology.

1 Introduction

The analysis of the shapes of objects has been of great interest in many areas,

such as medicine [6], materials science [16] and industrial processing [24,4]. In

fact, the measurement of shapes is an ongoing research topic, particularly in

digital image processing and discrete geometry [18,10,23]. Even though several

shape descriptors are useful for describing and diﬀerentiating a variety of objects

– circles, ellipses, rectangles, and reclines [22] – its computation is still a tough

task. The main reason is that many descriptors are aﬀected by factors such as

resolution in the representation, small irregularities in the contours (perimeter

inaccuracy) and noise. Furthermore, these factors are sensitive to diﬀerent as-

pects of the shape (for instance, regular or irregular shapes).

The majority of shape measures are focused on a two dimensional space and

are commonly represented in a plane or as images in which one of the most useful

R.P. Barneva et al. (Eds.): IWCIA 2012, LNCS 7655, pp. 75–89, 2012.

c

Springer-Verlag Berlin Heidelberg 2012

76 A.M. Herrera-Navarro, H. Jim´enez-Hern´andez, and I.R. Terol-Villalobos

measures is the circularity of the shape [24,21,19,8].The most widely used mea-

sure of circularity is the so-called shape factor (SF) given by SF =4πArea/P2

[3], where Pis the perimeter. This measure is deﬁned as the ratio of the sec-

tion area of an object to its perimeter. The digital equivalent of this circularity

measure was introduced in [13]. This measure has several drawbacks in prac-

tice: (i) it is not perfectly scale invariant, (ii) it is diﬃcult to interpret, (iii) it

is highly sensitive to small irregularities in the contours, (iv) it is dependent

on the resolution, and (iv) it is aﬀected by overlapping, rendering impossible

to characterize partially visible objects. These problems have motivated many

authors to propose new measures of circularity [19,13,1]. Recently, Ritter and

Cooper [21] reviewed and compared seven measures of circularity in terms of

resolution dependence, demonstrating that most of them are derivations of SF.

In their work, the authors proposed two new measures of roundness: the mean

roundness (MR) and the radius ratio (RR). These measures can be useful to as-

sess the circularity of regular objects in a two-dimensional space; however, when

the information of the object is given partially or there are abrupt changes in

the contours they become ineﬃcient.

As commented above, the diﬀerent approaches try to exploit geometrical in-

formation for measuring circularity. In summary, the approaches found in the

literature to compute the circularity can be divided in three main groups: Ap-

proaches based on the circular Hough transformation (HT) [9,7,26]; their main

disadvantage is the fact that computational and storage requirements of the

algorithm increase exponentially to the dimensionality of the curve.

Approaches based on the separation of the circle problem into discrete and

compu tat ional geome try [18,10,25,5,2]; however, these algorithms need to be ex-

tended to measure the extent of the deviation with a digital arc. Approaches

based on the reference shape [15,19,1]; which has the following drawbacks: the

generation of a digitized disk adds more complexity and it is necessary to know

the real object to generate a digitized disk according to the shape resolution.

Approaches based on circle ﬁtting [20,14,11]; in some cases these methods oﬀer

solutions with minimum error, but they are not necessarily the best solutions to

the data. In sum, the reader may ﬁnd in the literature several properties that

a good measure of circularity should have. One of the most representative is

the work of Haralick [13]. In his work, the author introduces four properties to

construct a good measure of circularity of closed ﬁgures: (i) the more a ﬁgure

becomes circular, the more the measure of its circularity increases, (ii) the values

for digital ﬁgures follow the values for the corresponding continuous ﬁgures, (iii)

the circularity measure is independent of the orientation, and (iv) the circularity

measure is area independent.

This paper focuses on introducing a new framework to measure circularity,

which is not aﬀected by overlapping, incompleteness of the borders, invariance of

the resolution, or accuracy of border detection, providing a good balance between

measure accuracy and the computational resources needed. This framework is

based on the generalization of the concept of disk in spaces with high dimension

under a certain induced norm. This yields a framework in which the roundness

A Probabilistic Measure of Circularity 77

Fig. 1. Theshapeofacircleinadiscretespace.Thecirclebecomesdeformedbythe

resolution, being hard to decide whether it represents a circle. The examples show

diﬀerent discrete circles with radius equal to: (a) 0, (b) 1, (c) 2, (d) 3, (e) 4, and (f)

10, given pixels.

is conceived by its properties instead of by the topology of a particular space,

thus allowing the deﬁnition and measurement of the roundness of objects in a

non-Euclidean space, which can be useful in a range of applications. Moreover,

experiments are conducted to demonstrate the eﬀects of the measure in a real

example. In particular, the case of a two dimensional image is analyzed and

matched with two circularity measures recently proposed.

This paper is organized as follows. In Section 2, the foundations of the cir-

cle and disk, together with their main properties, are introduced. In an initial

part, possible situations which could aﬀect the robustness of the measures are

discussed, followed by a short description of the norm. In Section 3, a new dis-

crete measure based on the distribution of the radius is proposed. In Section 4,

the proposal is tested under the Euclidean norm with other well-known accepted

circularity measures: radius ratio (RR) and mean roundness (MR). In additon, a

real example to compare these measures is presented. Subsequently, the results

are discussed showing the main advantages of our proposal with real digital

images. Finally, comments and conclusions can be found in the last section.

2 Foundations

In practical scenarios, the existent circularity measures become insuﬃcient be-

cause the discretization process of the image may deform the shape of the in-

volved objects as is appreciated in Fig. 1.

78 A.M. Herrera-Navarro, H. Jim´enez-Hern´andez, and I.R. Terol-Villalobos

2.1 Basic Notions and Preliminaries

The deﬁnition of a circle entails a center position cand a neighborhood of size

r. Assuming that cis in R2and r∈R+, the disk is deﬁned as:

O(c, r)={xi|d(c, xi)≤r}(1)

where d(c, xi)=(c1−xi1)2+(c2−xi2)2,r∈R+and c, xi∈R2. As a conse-

quence, the circle is deﬁned as follows:

C(c, r)={xi|d(c, xi)=r}.(2)

Both equations (1) and (2) represent inﬁnite sets 1under the corresponding

restrictions. However, this deﬁnition is limited to a two dimensional space, that

is, any measure based on these assessments is dependent on the dimensionality

of the space.

2.2 The Shape of the Circle and the Norm

In general terms, by preserving the deﬁnitions of circle and disk, we can substi-

tute the L2norm used to measure the distance by any Lknorm. As a result,

the shape and form of the circle changes, i.e., the topology of the neighborhood

is completely diﬀerent and depends on each norm. Then, the norm Lkdeﬁnes a

distance function dkas follows

dk(x, y)=n

i=1

|xi−yi|k

1

k

,(3)

where || is the absolute value.

3 Circularity Measure Framework

Whenever the circle is discretized, the pdf becomes aﬀected by the density of the

discretization, changing the form to a Gaussian-like form, where the μparameter

represents the radius and the σparameter the sparseness. These parameters

have a direct relation to the discretization process, i.e. f(r)≈G(μ, σ ). In other

words, when the circle becomes bigger the density function tends to a Gaussian.

However, environment perturbations of the discretization process cause that the

f(r) of the radius is composed of distinct peaks or modes. Therefore, the radius

distribution, without loss of generality, can be modeled as mixed distributions.

This model can become more complicated when there are insuﬃcient data to

estimate the set of parameters for each density function.

1Under the assumption that they are represented in a continuous space.

A Probabilistic Measure of Circularity 79

3.1 The Circularity Measure

Let us consider that the magnitude of the radius can be represented as a ran-

dom variable rwith a distribution f(r). Whenever the evidence of the elements

belonging to the circle Care the result of an acquisition process, they repre-

sent a subset of C(c, r), and the f(r) distribution becomes sparse. The expected

value E[f(r)] for f(r) corresponds to the most probable value for the radius. The

area surrounding the maximum of f(r) denotes the probability Pr(C), which

represents the circle C(c, r) of radius rand center c.

Pr(C)=b

a

f(r)dr (4)

such that C={x1,x

2,...,x

n}and xi∈Rn,whereaand brepresent the

interval of conﬁdence. The aand bvalues are deﬁned as E[f(r)] −g1(r, k)and

E[f(r)]+g2(r, k) respectively, such that functions g1and g2denote the maximum

sparse criterion.

Let C={x1,x

2,...,x

n}be such that each xi∈R2and dk(c, r)isatwo

dimensional norm kand c=(x, y) is the center, then the roundness measure is

deﬁned as:

MOR(C)=E[f(r)]+g2(r,k)

E[f(r)]−g1(r,k)

f(r)dr (5)

The proposed measure results invariant to the norm used, in the sense that it

follows the circle restriction regardless of the type of norm used.

3.2 Parameter Estimation

In some cases note that parameters cand rof the measure must be previously es-

timated to make a good measurement; however, in real scenarios, it is diﬃcult to

provide a good estimation of these parameters. To ﬁgure out the estimation task,

the measure of parameters will be considered in four main situations discussed

bellow:

1. The objects to be measured have closed borders. For a given border

denoted by C={x1,x

2,...,x

n},whereeachxi∈R2under the dknorm,

the center of the object is deﬁned as:

c=E{x1,x

2,...,x

n}(6)

and the radius distribution is deﬁned as follows:

r={dk(c, x1),d

k(c, x2),...,d

k(c, xn)},(7)

where the pdf of r∼f(r)

2. The objects to be measured contain partial information. Assuming

that the evidence of the border Ccan be represented as a diﬀerentiable

80 A.M. Herrera-Navarro, H. Jim´enez-Hern´andez, and I.R. Terol-Villalobos

parametrized curve in R2, the radius of circularity for the evidence Cis

estimated as follows

r=|γ|3

|γ|2|γ|2−(γ·γ )2(8)

such that γ

t=γt+1 −γt−1and γ

t=γ

t+1 −γ

t−1,whereγ(n)is expressed as

centered diﬀerences for the discrete case. Once the radius is estimated, the

center of the circle is located at a distance kfrom a vector orthogonal to its

derivative, which is denoted as follows

c=γ(t)+−γ(t)r

||γ(t)|| (9)

3. The border of the circle is not connected. The estimation of the center

is computed as the expected value of the estimated centers of all disconected

borders, i.e., for a given set of borders Ω={C

1,C

2,...,C

k}correspond-

ing to the same object, the estimated radius and centers are estimated as

described above and denoted as Φ={r1,r

2,...,r

k},Ψ={c1,c

2,...,c

k}.

Consequently, the radius of the semi arcs in Ωare deﬁned as:

E[Φ] = arg max{r1,r

2,...,r

k}(10)

4. The amount of information in the borders. The border of the object

may be closed or opened. In the ﬁrst case, the situation is quite similar to

the one mentioned above, with the consideration that a discretization process

involves a uniform sampling over the ideal border. However, when the cardi-

nality |r|is considerable, then to reduce the complexity of the computation,

f(r) can be estimated. Assuming a Gaussian parametric form, the estima-

tion may be performed by considering an optimization process, in which

the derivative of vector parameter θrof the distribution G(θ) is computed,

where θr=[μ, σ]. Then, the optimal parameters θr

∗can be estimated using

standard methods of diﬀerential calculus.

The parameter estimation is considered as a maximum-likelihood problem.

However, the nature of the data make it diﬃcult to guarantee the stability of

the parameter estimation. To overcome this problem, each evidence can be

considered as an incomplete estimation of the true parameters, where a func-

tion Q(θr,θ

∗

r) exists and represents the parameter estimated with incomplete

data θrand the true parameter θ∗

r. Iteratively, it is feasible to estimate the

value of θ∗

rfrom θrusing the following expression for the Gaussian parameter

estimation:

θt

r=ρθ(t−1)

r1+(1−ρ)x(t)

ρθ2(t−1)

r2+(1−ρ)(θ(t−1)

r1−x(t))2(11)

such that ρ[0,1]. This is a better way to estimate the parameters of the

distribution without the restriction of the loss of data.

A Probabilistic Measure of Circularity 81

The parameter estimation performed in the fourth case is suitable for the ﬁrst

and third cases, since it oﬀers an economic way to compute the parameter dis-

tribution of the center of the circle in a closed situation and the real center,

whenever partial information is presented.

4 Experimental Process

This section presents a set of experiments performed with the aim of showing the

capabilities of the proposal and its behavior against other well-accepted measures

in controlled and uncontrolled scenarios. The experimental process is divided in

three parts. In the ﬁrst part, the measure is tested under controlled situations,

one hundred digital circles are drawn, in which parameters such as norm and

radius are varied. Particular aspects such as resolution, occlusion, connectivity

and amount of border information of the objects are addressed here. In the

second part, the proposal is tested with other well-known measures – radius

ratio and mean roundness [21] – under the L2norm (Euclidean distance). In the

third part, real images of graphite nodules are used; the nodules are characterized

by their circularity using our approach and recent well-known used approaches.

4.1 Synthetic Images

In this subsection, the results obtained in controlled situations are presented.

They include the validation of the framework by evaluating the roundness mea-

sure when it is applied to various circles with diﬀerent norms and radii. The test

consists in applying the MOR measure over a set of ideal circles under diﬀerent

Linorms. The radius is varied from 1 to 100 with increments of 1, whereas the

norm corresponds to L0.001,L0.002,L0.005 ,L0.01,L0.02 ,L0.05 ,L0.1,L0.2,L0.5,

L1,L2,L3,L4,L5,andL10. Results are illustrated in Fig. 2. As can be appre-

ciated, the proposal obtains similar results for the diﬀerent norms. The lowest

measures correspond to norms L1<, where low values imply loss of accuracy for

the diﬀerent drawn circles. Additionally, in the majority of cases the measure is

located above 0.995 with circles of at least 15 pixels in radius (see next para-

graph for a more detailed explanation of this fact). On the other hand, several

norms with small radii are scored with values equal to 1. From this situation fol-

lows the sparseness process, which comprises those few data that do not provide

enough evidence for locating the inﬂexion change over the pdfs conformed by the

distance to the border. The independence of the framework over the norm used,

makes it suitable to be employed with other induced norm topologies in which

other forms such as squares, stars or even diamond shapes can be recognized

with a simple change in the norm used.

Next, in Fig. 3, an emphasis is made of the results obtained with the L2norm.

It can be observed that the MOR measure stabilizes around circles with 5-pixel

radius. This means that the information in the border provides enough evidence

to infer that the ﬁgure is a circle. This result is completely justiﬁed by the big

numbers law [12], which in general terms states that enough evidence to model

82 A.M. Herrera-Navarro, H. Jim´enez-Hern´andez, and I.R. Terol-Villalobos

Fig. 2. Results obtained from diﬀerent circles artiﬁcially generated with diﬀerent

norms. As can be appreciated, regardless of the norm used, the roundness measure

converges faster for values nearer to one.

Fig. 3. Results for the particular case L2. The circles with small radii converge faster

to one.

a normal distribution must consist of at least 30 elements. In our particular

case, the perimeter of a 5 pixel radius is 31.415 pixels. Similarly, circles smaller

than 5 pixels have a MOR measure approximately of 1 given by the limitations

of locating the inﬂexion points over the pdf of the radius. The above results

give the minimum criterion with respect to the resolution and the amount of

information necessary to infer that the ﬁgure represents a circle and provide an

issue to deﬁne a sampling process which reduces the computational complexity

to estimate the measure discussed in the fourth case.

The following test consists in evaluating the accuracy in the recognition of

a circle, even when there is a lack of information. This situation is common

when the object is partially occluded or the quality of the acquisition is poor

and the borders are not well-deﬁned. The test consists in sampling an artiﬁcial

circle using a percentage of the total of pixels that conform the border. The

A Probabilistic Measure of Circularity 83

Fig. 4. Results obtained for diﬀerent sampling levels. The degradation in the accuracy

of the MOR measure is small, which conﬁrms that it is robust, even when there is no

considerable information on circle borders.

radius is varied and the circle is used under the L2norm. The total of the

information used in the sampling is: 100%, 75%, 50%, 25%, 12.5%, 10%, 5%

and 1%. The results are illustrated in Fig. 4. The sampling process entails a

uniform sampling over the pixels that conform the border of the circle. The MOR

measure is not signiﬁcantly aﬀected by the sampling process. When the sampling

process uses only 10% or less of the information, there is a slight reduction in

precision; however, in practice this represents extreme cases that are not likely

to succeed. As can be appreciated, the proposal framework is robust even when

the information of the object is partial or has been lost.

In conclusion, the above test shows that the framework is suitable to measure

the roundness in various scenarios. In the following paragraphs we show the

results on real scenarios, as well as the comparison of our proposal with other

well accepted measures.

4.2 Comparison with Other Measures in Synthetic Images

In an attempt to validate the proposal, in this study, our measure is compared

with two other measures: the radius ratio (RR) and the mean roundness (MR).

RR =rbmin

rbmax

,(12)

where rbmin is the minimum radius from a border point to the center of the

border and rbmax is the maximum radius. The main disadvantage of this measure

is that the proportions of the shortest and longest radius do not provide suﬃcient

information to characterize the roundness of the object, and the measure can be

aﬀected by pixel aberrations as well.

84 A.M. Herrera-Navarro, H. Jim´enez-Hern´andez, and I.R. Terol-Villalobos

The second measure is based on the theory of mean deviation (MD), it calcu-

lates the sum of the absolute diﬀerences between the radius of each border pixel

and the average radius. This measure is deﬁned as:

MR =1

nrb

|rj−rb|+rb

,(13)

where rbis the average radius from the border points to the center of the object,

and rjis the radius of border point jto the center of the border. The center

of the object is usually expressed as the expected point from all points that

conform the border. This expression is valid only if the expected value is equal

to the mean. Therefore, the average is highly correlated for closed to round

objects. Consequently, this measure is not reliable when assessing the roundness

of irregular and partial shapes.

The results of the comparison are shown in Fig. 5. From this graph, it can

be seen that the proposed measure converges quickly for values near to one.

On the other hand, MOR and MR are equivalent when the resolution becomes

higher. Conversely, the RR measure yields signiﬁcant smaller results. This means

that it is less robust to resolution changes. Moreover, the main disadvantage of

this measure is that the proportions of the shortest and longest radii do not

provide suﬃcient information to measure the roundness and the measure could

be aﬀected by pixel aberrations. The MR measure becomes numerically similar

to the proposal; however, these results were obtained from regular shapes, but

when there is a partial or not connected shape (for instance an arc), it is not

reliable for measuring the roundness, because it is not possible to deﬁne the

rbradius. Furthermore, the mean does not always represent a good estimator

for characterizing the roundness of an object, especially when the objects to

be measured are partially occluded or the borders are not complete. This fact

becomes important when measuring the roundness of incomplete particles.

Commonly, several measures discuss the resolution as an invariance property

of each of them; however, as it was commented above, in Fig. 1, the shape and

the topology of the circle are related to the resolution and therefore unsuitable

for several measures that are considered resolution invariant.

Next, one hundred polygons were generated, the number of sides ranging

from 3 to 100. Figure 6 shows the measures obtained for regular polygons. As

expected, the circularity increases with the number of sides and converges toward

1. The larger the number of sides, the more the polygons resemble a circle and

the more the circularity approaches 1. Now, the RR measure is quite similar to

MOR whenever the polygons are regular; instead MR results less robust than

the other two. Hence, MR results more robust to resolution changes, but RR is

better to characterize regular polygons. However, our proposal appears always

equivalent to the best measure in the distinct tested scenarios.

4.3 Application to a Real Case

Finally, to show the reliability of the proposal, we have selected a real application

oriented to compound materials. The application corresponds to the analysis of

A Probabilistic Measure of Circularity 85

Fig. 5. Results obtained of diﬀerent circles artiﬁcially generated with diﬀerent measures

Fig. 6. Regular polygone circularity

nodule graphite circularity in images. The proposed approach is used under the

L2norm (Euclidean distance), and for comparative purposes the two measures

aforementioned are used. In Fig. 7, a typical image with graphite nodules is

shown. Note that the nodules come with diﬀerent shapes close to circular forms.

In order to test our approach, we develop an experimental analysis in two stages:

In the ﬁrst scenario, the order of each measure is deﬁned and analyzed over

10 distinctive graphite nodules (see Fig. 8). In a second scenario, 110 graphite

particles are evaluated in a graphite specimen and the range of distribution for

each measure is computed.

In the ﬁrst stage, from a population of 10 typical shapes encountered in the

nodules, the circularity measure of each shape is estimated. The image in Fig.

8 shows the distinctive graphite nodules; Table 1 displays the results for each

measure. Note that all measures have a distinguishable diﬀerence, especially

when the shape of the nodules is irregular. To illustrate this, observe the ﬁrst

element in the table, the minimum value of the roundness measure MOR is

0.4437, and those computed for MR and RR are 0.7563 and 0.0471, respectively.

86 A.M. Herrera-Navarro, H. Jim´enez-Hern´andez, and I.R. Terol-Villalobos

Fig. 7. Graphite specimen

Fig. 8. Diﬀerent graphite nodules

On the other hand, when the nodule shape tends to be more circular, MOR

and MR show similar values; this can be seen in samples 9 and 10 of Table

1. Analyzing these results with MR, the interval between the least roundness

and the most roundness is small, approximately 0.2130; consequently, it cannot

distinguish the eﬀects of roundness. Alternatively, taking our proposed measure

and RR, the interval is about 0.5314 and 0.8372, respectively, which means that

it is possible to discriminate between the least circular nodule and the most

circular one. On the other hand, to demonstrate the consistency of the measures

when compared with human perception, the 10 nodules were evaluated by 50

people. In this part, each person accommodated them in order from the most

circular shape to the least circular. The comparisons carried out between the

participants and the measures obtained show that MOR and MR give a good

match when human perception is utilized. A visual comparison evinces that not

all the selected graphite particles are close to a circle, that is, some nodules

cannot be considered circular. As a result, the MR measure is less accurate in

characterizing the circularity of a graphite sample, especially when the shape

is markedly irregular. Finally, as can be appreciated, our proposal maintains a

relative order of the shapes, being more robust for irregular objects, unlike the

other presented measures.

A Probabilistic Measure of Circularity 87

Table 1. Values of roundness estimated for 10 segmented nodules illustrated in Fig. 14

Nodule Number MOR MR RR

1 0.4737 0.7563 0.0471

2 0.5520 0.7615 0.1180

3 0.5520 0.6315 0.5381

4 0.5934 0.7748 0.4588

5 0.6330 0.7803 0.6147

6 0.6390 0.7765 0.5047

7 0.7361 0.9424 0.6315

8 0.9654 0.9358 0.6852

9 0.9851 0.9661 0.8216

10 0.9751 0.9693 0.8847

Fig. 9. Histograms for the diﬀerent measures

Finally, as an additional test, the sparseness of the measure is analyzed for

each measure used for nodules in the image (Fig. 7). Using all nodules, the dis-

tribution of each measure is estimated. Observe in Fig. 9 that the MOR measure

has the sparsest range. On the other hand, for MR and RR, the dispersion is less

sparse. In metric spaces, sparse distribution is better because the probability to

generate classes is higher. Then, when using MR and RR measures, it becomes

88 A.M. Herrera-Navarro, H. Jim´enez-Hern´andez, and I.R. Terol-Villalobos

more complicated to distinguish the diﬀerent classes of nodules. As aforemen-

tioned, the MR measure is based on the mean operator and MOR is based on the

mode. This diﬀerence causes the outliers to aﬀect the mean operator, whereas

the mode becomes more robust, especially when outliers do not follow a normal

distribution. It is noteworthy to emphasize that the proposed measure (MOR)

is a generalized measure. Therefore, it satisﬁes the following properties: (1) it

ranges within ([0,1]), where 1 is scored only by a perfect circle, (2) it is invariant

with respect to resolution, so the measure becomes independent of the equip-

ment, (3) it is tolerant to shape variations, (4) it is tolerant to noise or narrow

intrusions, and (5) it can be easily compared to human perception. Hence, the

proposed measure can be adopted as an alternative to measure the roundness of

graphite nodules.

5Conclusion

In this work, a new framework to measure the roundness is presented. This

framework is based on the central concept of circle in a two dimensional space

under certain induced norm. This framework results useful, because the round-

ness is conceived by its properties instead of by the topology of a particular

space, which allows to match order and to measure the roundness of the objects

in non-Euclidean spaces. The particular case of a two dimensional case is ana-

lyzed. Here, the proposed measure is matched with two of the most well accepted

measures. The results demonstrate that the proposed measure is better behaved

with regard to spareness and measures more robustly the roundness of the ob-

jects under the tested scenario. As a result, we provide a reliable framework to

deal with the task of measuring circularity.

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