A Probabilistic Measure of Circularity

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.

Full text

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 Care 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 Ccan 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 Cis
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
nrb
|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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