Euler Characteristic Computation by Means of a Chain Code Applied to Binary Images

Abstract

This paper presents a new approach for calculating the Euler characteristic in 2D binary images. The problem is addressed using the Three OrThogonal symbol chain code (3OT code), using only one symbol for the calculation of the Euler characteristic. Using this code, it is possible to introduce new geometric concepts represented by the same symbol of the 3OT alphabet and to simplify the overall equation of the Euler characteristic. This process is supported by the proof of a set of theorems and their numerical validation, using a set of binary images with a variable number of holes. Thus, this research proves that the 3OT code can be used not only for image compression as reported in the literature, but also to simplify the expression of the Euler characteristic as well as for the analysis and simplification of the shape of contours.

Full text

Euler Characteristic Computation by Means of a Chain
Code Applied to Binary Images
Elisa I. Gómez-Gómez
*
and Hermilo S
anchez-Cruz
†
Centro de Ciencias Basicas, Universidad Autonoma de Aguascalientes
Avenida Universidad 940, 20100 Aguascalientes, Ags., M
exico
*
elisa.gomez@edu.uaa.mx
†
hermilo.sanchez@edu.uaa.mx
Received 20 September 2023
Accepted 7 June 2024
Published 21 August 2024
This paper presents a new approach for calculating the Euler characteristic in 2D binary images.
The problem is addressed using the Three OrThogonal symbol chain code (3OT code), using
only one symbol for the calculation of the Euler characteristic. Using this code, it is possible to
introduce new geometric concepts represented by the same symbol of the 3OT alphabet and to
simplify the overall equation of the Euler characteristic. This process is supported by the proof
of a set of theorems and their numerical validation, using a set of binary images with a variable
number of holes. Thus, this research proves that the 3OT code can be used not only for image
compression as reported in the literature, but also to simplify the expression of the Euler
characteristic as well as for the analysis and simpli¯cation of the shape of contours.
Keywords: Euler characteristic; 3OT chain code; binary object; Parity Theorem; geometries in
the contour.
1. Introduction
A number of authors have used this characteristic to solve many problems, as in the
case of Refs. 26 and 34, where it is shown that this topological property is the most
suitable characteristic from the medical point of view to discriminate cervical dis-
orders. The interested reader can consult Refs. 1,2,10,11,18,19,40 and 44, for more
applications, patents and hardware implementations of the Euler characteristic.
Furthermore, this same property was successfully applied in Ref. 47 to extract
lung regions from grayscale chest X-ray images. Also in Ref. 15, the proposed ap-
proach was applied to two real-world examples. The ¯rst, the calculation of the
†
Corresponding author.
This is an Open Access article published by World Scienti¯c Publishing Company. It is distributed under
the terms of the Creative Commons Attribution 4.0 (CC BY) License, which permits use, distribution and
reproduction in any medium, provided the original work is properly cited.
OPEN ACCESS
International Journal of Pattern Recognition
and Arti¯cial Intelligence
Vol. 38, No. 10 (2024) 2454012 (36 pages)
#
.
cThe Author(s)
DOI: 10.1142/S0218001424540120
2454012-1

number of holes in a one-piece polyurethane slide; and the second, two examples
showing the calculation of the Euler characteristic applied to two images of indi-
vidual tumor gland segments from cribriform cases of prostate cancer. In Ref. 3,it
was used to automatically recognize the numbers and characters of Malaysian car
license plates. In Ref. 7, it has been used for gender discrimination from an Hindi
signature. The Euler feature has also been applied in digit recognition from pressure
sensor data, as described in Ref. 33. The same topological feature has been used in
the recognition of industrial parts, as reported in Ref. 47. It has even been exploited
to describe structural defects in binary images that have been a®ected by noise
(see Ref. 52).
Through the years, a vast number of researchers from all around the world have
ventured to develop new approaches to compute the Euler characteristic. Di®erent
techniques and representations have been employed for this purpose. For example,
Bin-trees,
8
Quadtrees,
20,37
Graph theory,
16,17,21,29
Object Skeletonization,
41
Contact
Perimeter,
14,42
the so-called family of Quermass integrals,
9
Morse operators,
32
run-
based algorithms
47
and algorithms for labeling connected components (CC).
27,28
Two techniques through which new Euler characteristic equations have been
derived by using arti¯cial intelligence techniques are Bit-Quad patterns
25
and Vertex
Chain Code (VCC).
12
In the case of bit-quad patterns, the relationship between
pixels has been observed to improve the popular algorithm proposed by Gray, which
consists of mapping a binary object and obtaining the frequency of occurrence of each
pattern of 2 2 pixels and thus obtaining its Euler characteristic.
25
Recently, in Ref. 23, an algorithm based on simulated annealing was designed to
¯nd optimal expressions for computing the Euler characteristic, considering the four
and eight connectivity. The proposed approach found the complete family of
expressions using three bit-quads patterns that correctly estimate this topological
property. In addition in this same work,
23
58 other new expressions using more than
three bit-quads were found.
Similarly in Ref. 6, authors propose to automatically derive the equations to
compute the Euler characteristic of a binary image. To do so, they train an Arti¯cial
Neural Network (ANN) to automatically ¯nd the optimal combinations of bit-quads to
compute the Euler characteristic of a binary image, pretending that these equations are
explainable.
In the case of the VCC,
12
which uses two vectors for encoding, it is invariant
under the a±ne transformations of rotation, translation and mirroring. In addition,
it is possible to represent shapes composed of triangular, rectangular and hexagonal
cells. Due to the way it encodes, it is possible to obtain the object perimeter, contact
perimeter, contact vertices and complementary chains of a two-dimensional (2D)
binary object to obtain new Euler characteristic equations. It has even been used to
train an ANN to calculate the Euler characteristic of a 2D binary image.
15,42,43
However, there is another technique such as the Three OrThogonal (3OT) chain
code that has not been used as a topological descriptor for the calculation of the Euler
E. I. G
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2454012-2

characteristic. This code is composed of three vectors: a support vector, a reference
vector and a change vector.
39
In addition, when chaining the object contour, sym-
bology is simpli¯ed for some geometries, as we prove in this work, which constitutes
an advantage regarding other chain codes. Moreover, it is invariant under transla-
tion, mirror and rotation transformations of a multiples of 90 . Furthermore, it has
demonstrated favorable outcomes in 2D image compression.
4,5,39,49–51
1.1. Contribution
In this paper, we present a Parity Theorem, which allows the detection of convex and
concave shapes in a general way in the contour of an object, as well as the de¯nition
of new geometries, the calculation of the number of holes in an object or binary image
and a new formulation of the Euler characteristic. In the process, we have achieved
new expressions that are easy to compute for both the number of vertices a binary
shape has and the total number of edges involved. We have also modi¯ed the Euler–
Poincar
eequation for 2D binary objects to consider di®erent neighborhoods of the
connected components, in which we were able to relate the discrete as well as the
continuous expression of the Euler characteristic. Finally, we validated the proposed
approach on a set of binary images and, even more, a simpli¯cation of a binary shape
and a classi¯cation of cells in the hippocampus region were also carried out.
1.2. Organization
This paper is organized as follows. In Sec. 2, we present concepts and de¯nitions used
throughout this work. Section 3presents the proposed approach and the calculation
of the Euler characteristic using the 3OT for binary objects with and without holes is
also described. Section 4presents some applications and results, a comparison of our
method with others known in literature, as well as the advantages of the proposed
method. Finally, in Sec. 5, conclusions and future work are presented.
2. Preliminaries
In order to introduce our approach and to show the relationship between the 3OT
chain code and the Euler characteristic, a number of concepts and de¯nitions are
described below.
The 3OT chain code has important features that allow describing a binary object
in terms of its Euler characteristic as it uses three vectors to encode its contour: one
called of reference, another one called of support and other of change.
38,39
The di-
rection changes are labeled by one of three symbols taken from the alphabet 3OT ¼
f0;1;2g(see Fig. 1), where the symbol 0 represents no change between the reference
vector and the support vector.
Since the 3OT chain code is designed to follow the pixel edges of the contour of a
binary object,
39
the changes made by the change vector with respect to the support
Euler Characteristic Computation by Means of a Chain Code
2454012-3

vector subtend right angles, resulting in convexities that can be encountered with an
angle of ¼
2and concavities with an angle of ¼
2.
Of course, the complete contour of a binary object can be composed of an external
contour, and, possibly, hole contours. Taking this into consideration, the external
contour is coded clockwise, while the hole contour is coded counterclockwise.
De¯nition 1. Encoding both, the external contour and the hole contour of an
object, starting with the upmost and leftmost pixel, as an initial condition for
convexity/concavity, the following statement is true. The ¯rst symbol 2 appeared in
a chain that represents an external contour is a 2^whereas the ¯rst symbol 2
appeared in a chain that represents its hole contour is a 2. The symbol 2, of the
external contour (corresponding to 2^), is chosen as the start of the chain code. See
Fig. 1.
This de¯nition is ful¯lled by the following assertion.
Assertion 1. When a circuit is completed in clockwise direction, under orthogonal
movements, the total sum of interior angles is 2, otherwise, when a circuit is
completed in counterclockwise direction, the total sum of interior angles is 2.
De¯nition 2. A string S3OT is a ¯nite ordered sequence of symbols from the set
3OT ¼f0;1;2gwhich is represented as S3OT ¼s1;s2;...;sn¼fsi:1ing,
where nrepresents the number of elements in the string.
Fig. 1. Example of a binary object encoded in terms of the chain code 3OT, clockwise for the external
contour and counterclockwise for the hole contour, with diagrams of the three symbols used to indicate a
change of direction in the object contour.
E. I. G
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De¯nition 3. A substring of S3OT , is represented as uand is considered as a
substring if and only if uappears within S3OT .
As an example of a string we have, S3OT ¼2201111111001111011011111101200
111101111000000210021002112120110211100112211111021011011221011
11110200121210110100001 2021011001102101102210110002100110111, which cor-
responds to the 3OT chain code of the contour shape (the blank space indicates that
the next part of the chain corresponds to a hole contour) of the binary object from
Fig. 1, then 2001111011110000002 is a substring of S3OT . In this paper, we work with
usubstrings starting with 2 and ending with 2. Such as, 200111101111000000X,
where Xrepresents the two-concave or two-convex symbol to be identi¯ed.
A string of 0's of the 3OT chain code, can be a sequence of zvectors in the same
direction, with z1
38,39
representing a plain (see (a) and (d) in Fig. 2), while symbol
1 presents an equal change to the reference vector (see Fig. 1), this symbol on the
contour generates a geometry of staircase (see (b) in Fig. 2) which is de¯ned as
follows.
De¯nition 4. Astaircase on the contour is denoted by a convexity followed by a
concavity, or vice versa, separated by none or more 0's symbols.
Symbol 2 represents a change in the opposite direction of the reference vector
39
(see (c) and (d) in Fig. 2), resulting in two important types of curvatures in the
contour, which help us to understand the way in which the calculation of our
proposed Euler equation is performed and are described in De¯nitions 5and 6.
Fig. 2. Geometries formed on the contour of a binary object using the 3OT chain code: (a) plain;
(b) staircase; (c) bump and (d) pothole.
Euler Characteristic Computation by Means of a Chain Code
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De¯nition 5. Given a bump, i.e. a protuberance on the contour formed by two
contiguous convexities separated by none or more symbols 0, where the ¯rst convexity
is labeled with a symbol 1 and the next with a symbol 2, that we denote it as 2^. We call
2^as the two-convex symbol of the 3OT code (see (c) in Fig. 2).
De¯nition 6. Given a pothole, i.e. a sinking on the contour formed by two contiguous
concavities separated by none or more symbols 0, where the ¯rst concavity is labeled
with a symbol 1 and the next with a symbol 2, that we denote it as 2. We call 2as the
two-concave symbol of the 3OT code (see (d) in Fig. 2).
Also, from a starting point the 3OT chain code can be handled in two directions,
clockwise and counterclockwise. Depending one the direction, it should be assigned
symbols 2 on di®erent positions on the contour shape.
De¯nition 7. Let 0þ1þ2þþnþ00 be the sum total of the angles
subtended by a substring ustarting and ending in 2 with w1's between them, such
that 0,00 and i,withi¼1;...;n, indicates orthogonal movements, i.e. 0,00 and
i2f=2;=2g.
De¯nition 8. Let be the angle subtended at the point of change, in which a
symbol 2 is assigned. If there is a convexity, i.e. ¼
2, we say that the symbol 2 is a
two-convex, or a 2^, for short. On the contrary, if there is a concavity, i.e. ¼
2, then
there is a two-concavity, or a 2, for short.
De¯nition 9. If we have a sum 0þ1þþnþ00 , it is satis¯ed that
nþ00 ¼; n¼=2;
; n¼=2:
ð1Þ
De¯nition 10. In this work, we consider break points to be the pixels of the contour
shape that we identify when a symbol 2 appears at one or more vertices of those
pixels.
2.1. Image triangulation
One of the most interesting topological properties of a binary object contained in a
2D image, Iðx;yÞ2f0;1g, is the Euler characteristic, which gives us a global de-
scription of the shape of the object, and also, it is a property that is not a®ected by
any deformation since it is invariant under linear and nonlinear geometric trans-
formations of the image, as long as there are no tears or joins in the object.
9,23,24
Starting from a binary object that is triangulated into su±ciently small cells, this
property was de¯ned in terms of its local characteristics called simplexes, i.e. the
total number of vertices ðn0Þ, line segments ðn1Þand 1's regions formed from the
resolution cells ðn2Þ
25
as de¯ned as in Eq. (2).
E1¼n0n1þn2:ð2Þ
E. I. G
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omez & H. S
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2454012-6

However, since the complete image consists of CC regions of 1's called connected
components and Hregions of 0's belonging to the holes,
27
Euler characteristic has
also been de¯ned as in Eq. (3).
E¼CC H:ð3Þ
For an object without holes and counting each simplex to obtain such a topological
property using Eq. (2), one notices that its Euler characteristic does not change, it is
always E1¼1, regardless of the neighborhood of each pixel (see Figs. 3(b) and 3(d)),
and Eqs. (2) and (3) are equal. However, a ¯nite set of pixels simply connected with
a certain number of holes arbitrarily distributed among the object, can produce
di®erent Euler characteristics (see Figs. 3(a) and 3(c)).
As is well known in the literature, there is a connectivity relationship between
each pixel, of a 2D binary object, either because they share vertices, edges or faces.
These relationships have been centralized into two types of neighborhoods that are
de¯ned as follows:
N4ðpÞ¼fðx;yÞ;ðxþ1;yÞ;ðx1;yÞ;ðx;yþ1Þ;ðx;y1Þg ð4Þ
and
N8ðpÞ¼N4ðpÞ[fðxþ1;yþ1Þ;ðxþ1;y1Þ;ðx1;yþ1Þ;ðx1;y1Þg:ð5Þ
Something that we must highlight in this work is that Eqs. (2)and(3)arethesameinthe
following cases. If we consider the CC in the N8neighborhood, then the holes should be
considered four-connected. That is, the connections of the connected one-pixel compo-
nents do not allow the connected zero-pixel components (i.e. the holes) to also be eight-
connected. Equating (2)and(3) fails when we want to consider both the CC and the
holes as eight-connected components at the same time (see Figs. 3(a) and 3(c)). Thesame
behavior happens when both, the CC and the holes, are considered four-connected.
Consequently, the computation of the Euler characteristic depends on the
neighborhood of the pixels that is applied. An example of such a statement is given in
(a) (b) (c) (d)
Fig. 3. Di®erent triangulations of one binary object: (a) and (c) with holes; (b) and (d) without holes.
Euler Characteristic Computation by Means of a Chain Code
2454012-7

Fig. 3(d), where we have the following values when counting their simplexes,
n0¼72, n1¼115 and n2¼44, employing Eq. (2) its Euler characteristic is
E1¼72 115 þ44 ¼1, the result of such determination is the Euler number cor-
responding to consider the object as a single component, which is equivalent to use the
N8neighborhood, a simple way to verify it is using Eq. (3) where it is known that the
number of connected components is CC ¼1 and the number of holes inside is H¼0,
then E¼10¼1. On the other hand, if we want to work with a N4neighborhood, we
have three connected components, then, CC ¼3 and no hole: H¼0, using Eq. (3)we
know that E¼30¼3. However, if we use the Eq. (2) we have that n0¼72, n1¼
115 and n2¼44; therefore, E1¼1. Thus, Eqs. (3) and (2) do not match.
2.2. Open problems of Euler characteristic computation
Equation (2) is valid for objects under eight connectivity but invalid for four-con-
nectivity. It is di±cult to estimate the Euler characteristic using this neighborhood
and Eq. (2) since it is not possible to determine the number of connected components
separately by local property measurements, as this equation was designed for dis-
cretized sets, without taking into account that a 2D image can have arbitrarily large
and intertwined objects and holes.
25
In addition to the fact that, when considering
neighborhoods in the binary object, N4increases the number of connected compo-
nents present in the object. Unlike Eq. (3) which refers to the continuous case, i.e. to
compact and connected subsets of R2, leaving open whether this equation refers to a
connected set or to several connected sets.
In order to match Eqs. (2) and (3), we add another parameter in Eq. (2). One way
to solve this problem is to consider that N4breaks with the topology of the object, i.e.
it divides the set of one-pixels in small entities and only connected laterally, in this
sense, a solution is to count separately the simplexes of each component generated by
N4, a more e®ective solution would be to consider as a local property the number of
times a vertex connects the binary object in N8denoted as n, as shown below.
E¼n0n1þn2þn;ð6Þ
where nis equal to the number of vertices connecting four-connected components in
its eight-vicinity. In Appendix A is demonstrated Eq. (6) by induction.
To numerically check Eq. (6), let us use the object shown in Fig. 4. For one hand,
let us consider Fig. 4(a) where the connected component (CC ¼1) is in N8and the
hole (H¼1) in N4. Using Eq. (3), E¼11¼0, whereas using Eq. (2), where
n0¼36, n1¼55 and n2¼19, we have E1¼36 55 þ19 ¼0, and both equations
are equal. Now, if we work with N4we have two connected components, i.e. CC ¼2
and no hole (zero-pixels), since the one-pixels that delimit the hole are not four-
connected (see the red points in Fig. 4(b)), therefore, H¼0. Now, if we employ
Eq. (3) then E¼20¼2. On the other hand, we can observe there are two
connections to vertices in N8(red points), so, n¼2. Using the proposed Eq. (6)
to obtain the Euler characteristic, E¼36 55 þ19 þ2¼2, as expected.
E. I. G
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In Fig. 5are presented other interesting examples. For these objects, consider only
one connected component (CC ¼1) in N8and two holes in N4to equate Eqs. (2)
and (3), as is usually done in literature, we have E¼1 for both equations.
On the contrary, let us now consider connected components in N4whereas their
holes in N8. Figure 5(a) presents an object with n0¼23;n1¼36 and n2¼12. If only
one connected component in N4and one hole in N8are considered, then E¼11¼0
for Eq. (3). However, since there is one connection in its N8hole, we have n¼1, so,
using the proposed Eq. (6) we have E¼23 36 þ12 þ1¼0, as expected.
On the other hand, Fig. 5(b) has an object with n0¼21, n1¼32 and n2¼10.
Note that there are two connected components in N4and no hole is present, since the
whole zero-pixels are part of the background in N8. So, E¼20¼2, according to
Eq. (3). Now, considering Eq. (6), in which n¼3 (three red points), we have:
E¼21 32 þ10 þ3¼2, as expected.
Finally, consider Fig. 5(c) that has an object with n0¼17, n1¼24 and n2¼6. In
this case, observe there are six connected components in N4and no hole in N8,
therefore, E¼60¼6 using Eq. (3) and if we use Eq. (6), where n¼7
(red points), we have E¼17 24 þ6þ7¼6, as expected.
(a) (b) (c)
Fig. 5. Three objects with E1¼1 (a) n0¼23, n1¼36 and n2¼12; (b) n0¼21, n1¼32 and n2¼10
and (c) n0¼17, n1¼24 and n2¼6.
(a) (b)
Fig. 4. Connected components with n0¼36;n1¼55;n2¼19, E1¼0 in: (a) N8, where n¼0 and
E¼0, and (b) N4, where n¼2 and E¼2.
Euler Characteristic Computation by Means of a Chain Code
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It can be concluded that the determination of the Euler characteristic depends
speci¯cally on the neighborhood of the pixels and hence on the choice of the neigh-
borhood either N4ðpÞor N8ðpÞ.
Summarizing this section, to equate Eqs. (2)and(3), part of this work is to focus
on making a correction on E1by n, in such a way Eqs. (2) and (6) are now possible
to equate, due to neighborhood to be considered. Original Euler–Poincar
eequation is
currently valid for eight-connected components, however we generalized for both;
eight and four connected by introducing n, whose value is zero for eight-connectivity
and di®erent than zero for four-connectivity. See, for example, Figs. 3–5.
In the following section, we describe some methods to propose new equations of
the Euler characteristic, as well as, methods to improve the existing equations in
order to reduce their complexity and also, we describe some applications that have
been given to such an important topological descriptor.
3. Proposed Approach
In order to calculate the Euler characteristic based on the 3OT chain code, it is
necessary to know the number of concavities and convexities, represented by the
symbol 2, using the S3OT chain of a binary object. From De¯nition 1, we know that
the ¯rst symbol 2 of a 3OT code always represents a convexity, whereas a concavity if
it is a hole. To ¯nd out what the rest of the symbols 2 represent, we prove the
following theorem.
Theorem 1. Let w be the number of symbols 1's separating two symbols 2in the
chain S3OT and let siand sjbe the ¯rst and the last symbol 2, respectively,of a
substring in S3OT.Then
(a) If sirepresents a convexity ðconcavityÞ;sjrepresents the same convexity
ðconcavityÞas if and only if w is even.
(b) If sirepresents a concavity ðconvexityÞ;sjrepresents a convexity ðconcavityÞif
and only if w is odd.
Proof. Conditions:
(1) When there is a symbol 2 with an ¼
2angle, the symbol 1 following it
subtends ¼
2(it is of opposite sign).
(2) When there is a vector v1in the opposite direction of another vector v2, the
cumulative sum of the angles is P¼, in the chain connecting v1with v2,
is satis¯ed.
Basis step (nis odd)
(a) 2^followed by a 1 ðn¼1Þ. We have 2^1X. How is X?
As symbol 1 subtends 1¼
2, since 2^subtends 
2, then the next symbol 2 is 2
because it subtends 00 ¼
2since 1þ00 , because the vectors associated
with 1 and 2are in opposite directions and make a cumulative ¼1þ00 ¼.
E. I. G
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2454012-10

(b) 2followed by a 1 ðn¼1Þ. We have 21X. How is X?
Since, we have 0þ1þ00 it must be ful¯lled that 1þ00 ¼, by Condi-
tion 2. But 1¼
2since 1 represents change in the same direction and sense.
Therefore, 
2þ00 ¼)00 ¼
2. By De¯nition 8,X¼2^.
Basis step (nis even)
(a) 2^followed by a pair of 1's. We have 2^11X. How is X?
As we have 0þ1þ2þ00 where 2¼
2, then by De¯nition 9and
Condition 2 it must ful¯ll that 2þ00 ¼, then, since 2¼
2,
2þ00 ¼
)00 ¼
2. Therefore, X¼2^.
(b) 2followed by a pair of 1's. We have 211X. How is X?
As we have 0þ1þ2þ00 where 2¼
2, then by De¯nition 9and Condition
2 it must ful¯ll that 2¼
2,
2þ00 ¼)00 ¼
2. Therefore, X¼2.
Induction hypothesis. Assuming it is valid for n. To prove for nþ1 (odd)
(a) 2^followed by nþ1 1's: 2^11X.Wehave0þ1þþnþnþ1þ00.
Since 0¼
2and nis even, then 1þ þn¼0. Since the theorem is valid
for n, then n¼
2)nþ1¼
2)
2þ00 ¼)00 ¼
2)X¼2.
(b) 2followed by nþ1 1's: 211X. We have 0þ1þþnþnþ1þ00 .
Since 0¼
2and the sum 1þþnþnþ1is with an even number of terms,
then 1þþn¼0 where n¼
2)nþ1¼
2)
2þ00 ¼)00 ¼

2)X¼2^.
To prove for nþ1 (even)
(a) 2^followed by nþ1 1's: 2^11X:We have 0þ1þþnþnþ1þ00.
Since 0¼
2and nis odd, then 1þþn¼
2. Since the theorem is valid
for n, then n¼
2)nþ1¼
2)
2þ00 ¼)00 ¼
2)X¼2^.
(b) 2followed by nþ1 1's: 211X. We have 0þ1þþnþnþ1þ00 .
Since 0¼
2and the sum 1þþnþnþ1is with an odd number of terms,
then 1þþn¼
2where n¼
2)nþ1¼
2)
2þ00 ¼)00 ¼

2)X¼2.
To numerically check Theorem 1, we use the set of binary objects shown in Fig. 6.
The 3OT chain code of a binary object is analyzed as a ribbon, where its beginning
and end are joined. Each symbol assigned to the object contour is dependent on the
previous symbol. Starting with Fig. 6(a) we can visually check that the ¯rst symbol 2
of the string marked on the contour is a convexity. Theorem 1works with substrings,
so s1is now the ¯rst symbol 2 of the substring to be analyzed (see Fig. 6(b) red
point), s8the next symbol 2 to be distinguished (see Fig. 6(b) blue point) and
u¼21111112 the substring to which they belong. We can easily check that the
number of 1's separating them is an even number, w¼6, so, by Theorem 1(a), s8¼2
is a convexity. Now, s8is the next reference point (see Fig. 6(c) red point). The next
symbol 2 in the chain is at position s14 in the substring u¼2101112. Since the
Euler Characteristic Computation by Means of a Chain Code
2454012-11

number of symbols 1's is also even, therefore, s14 is a convexity (see Fig. 6(c)). The
same is true for the substring in Fig. 6(d) knowing that now s14 the next reference
point is a convexity and that wis even, s17 likewise represents a convexity at that
point on the binary object contour. Figures 6(b)–6(d) show that as long as wis an
even number and the reference symbol 2 is a convexity, the subsequent symbol 2 is
also a convexity regardless of how large wis.
Fig. 6. Numerical veri¯cation of Theorem 1: (a) The object is covered by the 3OT chain code; (b)–(d) and
(h)–(i) convexities found from a convexity with even w; (e) concavity found from a convexity with odd w;
(f) concavity found from a concavity with even w; (g) convexity found from a concavity with odd w;
(j) concavities and convexities.
E. I. G
omez-G
omez & H. S
anchez-Cruz
2454012-12

Next, in Fig. 6(e) the opposite case occurs, we have in s17 a convexity and the
value of wup to the position of the next symbol 2, an odd number, in this case w¼1,
consequently the symbol 2 of the position s20 represents a concavity. Up to this point
the ¯rst case of Theorem 1(a) and the second case of Theorem 1(b) are satis¯ed.
Now, in Fig. 6(f) we have a value of wequal to 0, in this theorem the number 0 is
considered an even number, therefore, the second case of Theorem 1(a) is ful¯lled, in
which, the symbol of position s20 represents a concavity and wan even number,
consequently, the symbol of position s21 represents a concavity.
Once s21 has been distinguished as a concavity, it is now taken as a reference point
(see Fig. 6(g)) to characterize the symbol 2 of the position s23 of the substring
u¼212, since the number of 1's symbols separating each symbol 2 is odd w¼1, the
¯rst case of Theorem 1(b) is satis¯ed, s23 is a convexity. So, up to this instance, all
four possible ways of ¯nding a convexity and a concavity in the contour of a binary
object are satis¯ed. This numerically validates Theorem 1.
Moreover, from the convexity found at position s27 of Fig. 6(h) (blue point), it can
be numerically veri¯ed in Fig. 6(i) that the symbol 2 of position s1initially consid-
ered as a convexity indeed is, since in its substring u¼21102 the value of wis even,
the ¯rst case of Theorem 1(a) is then satis¯ed.
In Fig. 6(j), we can see all the convexities or bumps (green color) and all the
concavities or potholes (yellow color) found in the contour of a binary object from the
proposed Parity Theorem 1. As a remark, this numerical check is also valid for the
holes contours, just considering that the direction of the coding is counterclockwise.
3.1. Calculation of the Euler characteristic using the 3OT chain code
Theorem 2. Let N2^and N2be the number of convexities and concavities,
respectively,represented by the symbol 2, that are added in the contour.The number
of added holes of the binary object is given by
H¼N2^N2
4:ð7Þ
Proof. A hole is generated when N2^N2¼4. That is, if Hholes are
added, then
N2^N2¼4H:ð8Þ
Then
H¼N2^N2
4:ð9Þ
Considering Fig. 7(a) it can be appreciated that the object has N2^¼6 and
N2¼2, when a handle is added (see Fig. 7(b)), four more concavities appear,
i.e. N2¼4, whereas the number of convexities remains the same as before, i.e.
N2^¼0. Therefore, H¼1, i.e. one hole is computed. This numerically validates
Theorem 2.
Euler Characteristic Computation by Means of a Chain Code
2454012-13

Theorem 3. Let N2^be the number of convexities and N2be the number of
concavities,represented by the symbol 2. The number of holes of any binary object is
always given by
H¼N2^N2
4þ1:ð10Þ
Proof. For the base case, starting from a minimal binary object, in this case
composed of one pixel, then N2^
kN2
k
4¼1, where the su±x kstands for initial. Then, if one
or more pixels are added to such initial shape, and as long as no new hole is produced,
it is easy to verify that N2^N2
4¼1. Let us do the proof by induction of the number of
holes. Thus, if we continuously add pixels to the initial shape until a new hole
appears, and following Theorem 2, we obtain
H¼N2^N2
4¼H:ð11Þ
However, the total number of convexities of the binary object with a hole is given by,
the sum of the total convexities of the initial shape plus the convexities of the shape
of the hole:
N2^¼N2^
kþN2^ð12Þ
and the total number of concavities of the binary object with a hole is given by the
sum of the concavities of the initial shape plus the concavities of the hole shape:
N2¼N2
kþN2ð13Þ
then by substituting N2^and N2from Eqs. (12) and (13), respectively, into
Eq. (11) we obtain
H¼
N2^N2^
kN2þN2
k
4
(a) (b)
Fig. 7. Binary objects: (a) object and (b) object with a handle.
E. I. G
omez-G
omez & H. S
anchez-Cruz
2454012-14

¼
N2^N2
4þN2^
kN2
k
4
¼
N2^N2
4þ1:ð14Þ
When a binary object has only one hole, we have N2^¼4 and N2¼4 (see
Fig. 8(a)), then using Eq. (10), H¼1. With two holes, we have (see Fig. 8(d))
N2^¼5andN2¼9 substituting in Eq. (10), H¼59
4þ1 therefore H¼2. We
see that the formula holds even if we keep increasing the number of holes in a binary
form, such is the case in Fig. 8(b) with N2^¼6 and N2¼14 substituting in
Eq. (10), H¼614
4þ1 therefore H¼3, which is true.
Corollary 1. For any binary object,if the number of convexities is equal to the
number of concavities,i.e.N2^¼N2,then the binary form has only one hole.
Proof. This is demonstrated by direct calculation using Eq. (10).
(a) (b)
(c) (d)
Fig. 8. Types of holes: (a) and (d) holes with concavities; (b) holes with concavities and convexities; (c)
binary object without holes.
Euler Characteristic Computation by Means of a Chain Code
2454012-15

Theorem 4. Let N2^be the number of convexities and N2be the number of
concavities,represented by the symbol 2. The number of holes of any binary objects is
always given by
H¼N2^N2
4þCC;ð15Þ
where CC corresponds to the number of connected components.
Proof. Starting from any number of connected components then N2k
^N2k

4¼CC by
adding N2^and N2we then have that
H¼H
¼
N2^N2
4
¼
N2^N2^
kN2N2
k
4
¼
N2^N2
4þN2^
kN2
k
4
¼
N2^N2
4þCC:ð16Þ
The numerical check of this theorem is quite similar to Theorem 3, only considering
that the number of connected components is no longer limited to 1, it can be a
discrete number of binary objects present in a 2D image.
It is also possible to calculate the number of one-pixel regions (without holes) con-
tained in a hole, but, this is valid only for proper holes, i.e. a zero-pixel component
connected under the N4which Refs. 15 and 36 have de¯ned as a proper hole. Then
Theorem 5. The number of one-pixel regions without holes in any proper hole is
obtained as follows:
CC ¼N2N2^
4þ1:ð17Þ
Proof. See the proof of Theorem 4. To validate Theorem 5numerically, it can be
seen in Fig. 9that there are four connected components inside the hole where N2^¼
20 and N2¼8. Using Eq. (17) to verify this, we have that CC ¼N2N2^
4þ1¼
820
4þ1¼4 as expected. This numerically validates Theorem 5.
Theorem 6. For a binary object,the Euler characteristic can be calculated by the
following equation:
E¼N2^N2
4:ð18Þ
For any binary object, if the number of convexities N2^is equal to the number of
concavities N2, i.e. N2^¼N2, the Euler characteristic is equal to 0. Considering the
E. I. G
omez-G
omez & H. S
anchez-Cruz
2454012-16

Fig. 8(a) with N2^¼4 and N2¼4 and Eq. (18), we have that 44
4¼0. Which is
true, because if we use Eq. (10), knowing that we have one connected component
H¼44
4þ1, then H¼1, substituting in Eq. (3), E¼11, therefore E¼0.
Proof. Starting with Euler's equation and Theorem 6
E¼CC H
¼CC 
N2^N2
4þCC

¼N2^N2
4:ð19Þ
To validate Theorem 6, we use Fig. 8(c). As we can see, the Euler characteristic is
equal to 1. Verifying this, Theorem 8 is applied and it is obtained that N2^¼5 and
N2¼1and51
4¼1.
The proposed method has the ability to be applied both in objects connected
under N4and N8, this can be appreciated in Figs. 10(a)–10(c), where there are three
binary objects described in terms of the 3OT chain code connected under neigh-
borhoods 4 and 8. Applying the Parity Theorem 1we can distinguish 18 2^convex
symbols and 14 2concave symbols, applying Eq. (19) from Theorem 6we have that
E¼1814
4, therefore E¼1, which is true. Also, knowing that there are three con-
nected components in the image, applying Eq. (15) from Theorem 4we have that
H¼1814
4þ3, therefore H¼2, which is also true.
Speci¯cally in Fig. 10, it can be observed how N4(see Fig. 10(a)) and 8
(see Fig. 10(b)) are applied under a binary object. In that sense, each component is
Fig. 9. Numerical veri¯cation of Theorem 5: Hole with four objects inside.
Euler Characteristic Computation by Means of a Chain Code
2454012-17

encoded and its Euler characteristic is obtained in both neighborhoods. This tells us
that the method is indeed able to be applied in both neighborhoods.
3.2. Euler characteristic, 22neighborhoods and simplexes relationships
Let us remember that the Euler characteristic is given by a linear dependence of the
simplexes, that is, of the vertices, edges and resolution cells (see Eq. (2)). Let us
suppose, also, that there is a linear relationship between simplexes and arrays of
22 neighborhoods. Let us call these arrays as tetrapixels, such that the amount,
Nt, is given by Eq. (20):
Nt¼n0x0þn1x1þn2x2þnx;ð20Þ
Fig. 10. Neighborhoods: (a)–(c) four and eight adjacency with holes and three binary objects; (d) Euler
characteristic of the binary object applying eight connectivity; (e) Euler characteristic of the binary object
applying four connectivity.
E. I. G
omez-G
omez & H. S
anchez-Cruz
2454012-18

where xiare the scalar factors to look for, n0;...;n2are the simplexes and nis the
number of times that a vertex connect the four-connected components in its eight-
neighborhood.
Obviously, four adjacent pixels are required to form a tetrapixel. Figure 11 shows
some pixel arrays, where Fig. 11(d) consists of one tetrapixel but the others,
Figs. 11(a)–11(c), do not.
Let us use, for example, Figs. 11(a), 11(b) and 11(d) that satisfy the equations system
4x0þ4x1þx2þ0x¼0;
7x0þ8x1þ2x2þx¼0;
8x0þ10x1þ3x2þ0x¼0;
9x0þ12x1þ4x2þ0x¼1;
9
>
>
=
>
>
;
ð21Þ
44100
78210
810300
912401
0
B
B
@
1
C
C
A
:ð22Þ
Solving the extended matrix (22) by means of Gauss–Jordan, we obtain an expression for
number of tetrapixels given by the following equation:
Nt¼n02n1þ4n2þn:ð23Þ
In Appendix A is demonstrated Eq. (23) by induction.
Clearly, computing simplexes is more cumbersome than computing 2 2 neigh-
borhoods. To do this, we consider the relationship that exists between the contour
perimeter (easy to compute by using 2 2 pixel arrays). First, let us consider
Eq. (24), that give us the relationship between contact perimeter, Pc, and contour
perimeter, P, which is valid for four- and eight-adjacency.
14
2PcþP¼4n2;ð24Þ
where we are using n2as the number of pixels.
On the other hand, since the total number of edges of a binary object represented
by resolution cells is given by the following equation:
n1¼PcþP;ð25Þ
(a) (b) (c) (d)
Fig. 11. (a) one pixel, (b) two adjacent pixels, (c) three adjacent pixels and (d) a tetrapixel.
Euler Characteristic Computation by Means of a Chain Code
2454012-19

combining (24)with(25) we can conclude that number of edges is given by the
following equation:
n1¼2n2þ1
2P:ð26Þ
To achieve a simple expression for n0, we can use Eqs. (23)–(26) and obtain the
following equation:
n0¼NtþPn:ð27Þ
Therefore, we have achieved easy-to-compute expressions for both the number of
vertices that a binary shape has and the total number of edges involved, i.e. the
number of edges is twice the number of pixels plus half the perimeter, while that the
total number of vertices is the number of tetrapixels plus the perimeter of the contour
minus the number of eight adjacency connections.
As an example, Fig. 12 shows the validity of the formulas we found.
Therefore, we can ensure the following: each chain code faithfully contains the
information of the object's contour, but in turn, each contour is related to the interior
of the object, which at the same time, both the interior and the perimeter contain the
information of what happens in the neighborhood of 2 2 pixels.
4. Application and Results
In order to illustrate the capabilities of the proposed method, we present three
di®erent applications: ¯rst, the calculation of the Euler characteristic and the
number of holes of a set of binary images with di®erent characteristics; second, the
simpli¯cation of the binary shape of an object, and third, the classi¯cation of neu-
ronal cells using histological images in terms of the two-concave and two-convex
symbols.
(a) (b)
Fig. 12. Examples for checking our formulas for an object (a) without holes and (b) with holes.
E. I. G
omez-G
omez & H. S
anchez-Cruz
2454012-20

4.1. MPEG-7 dataset
In order to show the robustness of the proposed method, a set of 74 images of
di®erent sizes, shapes, contour noise, and a variable number of holes were selected
from the MPEG-7 dataset.
35
These images were described in terms of the 3OT chain
code, along with their respective holes. Subsequently, Theorem 1was applied to the
resulting chains of each image, and its Euler characteristics and the number of holes
were calculated using Eq. (18) of Theorem 6and Eq. (10) of Theorem 3, respectively.
The experimental results are summarized in Table 1. It can be observed that, in all
cases, the Euler characteristic and the number of holes are correctly computed, as
Table 1. Results obtained for the Euler characteristic with the proposed approach for the sample of
objects.
Image Size N2^N2Holes Euler Image Size N2^N2Holes Euler
Apple 256 256 11 7 0 1 Fish 284 127 39 35 0 1
Bat 851 856 268 280 4 3 Flat¯sh 613 492 42 38 0 1
Beetle 313 319 169 201 9 8 Fly 270 200 72 168 25 24
Bell 252 255 21 17 0 1 Fork 640 480 30 26 0 1
Bird 418 617 27 291 67 66 Fountain 360 243 7 3 0 1
Bone 361 441 11 7 0 1 Frog 198 175 41 197 40 39
Bottle 352 288 5 1 0 1 Glas 335 374 8 4 0 1
Brick 352 288 10 6 0 1 Guitar1 299 200 11 15 2 1
Butter°y 518 336 52 164 29 28 Guitar2 325 148 17 65 13 12
Camel 390 339 34 30 0 1 Guitar3 402 702 55 291 60 59
Car 352 288 6 12 0 1 Hammer 720 371 10 6 0 1
Carriage 352 288 12 8 0 1 Hat 158 152 12 80 18 17
Cattle 445 275 130 322 49 48 HCircle 203 337 6 2 0 1
Cattle1 536 383 121 477 90 89 Heart 471 390 7 3 0 1
Cellular 426 175 11 7 0 1 Horse 315 266 58 122 17 16
Chicken 283 246 43 91 13 12 Horseshoe 182 216 97 125 8 7
Children 352 240 13 9 0 1 Jar 512 512 12 12 1 0
Chopper 352 288 16 12 0 1 Key 406 259 10 6 0 1
Classic 1111 491 46 42 0 1 Lizzard 538 774 126 174 13 12
Comma 284 446 6 2 0 1 Lm¯sh 660 658 57 53 0 1
Crown 376 204 16 24 3 2 Misk 354 417 14 10 0 1
Cup 343 296 9 5 0 1 Octopus 256 256 32 28 0 1
Deer1 533 652 79 367 73 72 Pencil 487 186 8 4 0 1
Deer2 660 584 100 184 22 21 Personal car 549 237 16 12 0 1
Device 0 512 512 179 175 0 1 Pocket 366 536 73 477 102 101
Device 1 512 512 200 196 0 1 Rat 352 288 20 16 0 1
Device 2 558 558 148 144 0 1 Ray 635 638 240 384 37 36
Device 3 558 558 224 220 0 1 Sea snake 378 344 25 21 0 1
Device 4 558 558 135 131 0 1 Shoe 404 255 11 7 0 1
Device 5 558 558 99 95 0 1 Spoon 256 256 6 2 0 1
Device 6 558 558 24 20 0 1 Spring 640 480 102 98 0 1
Device 7 512 512 45 41 0 1 Stef 352 240 8 4 0 1
Device 8 558 558 21 17 0 1 Teddy 360 243 17 13 0 1
Device 9 558 558 61 57 0 1 Tree 425 394 26 22 0 1
Dog 528 409 131 171 11 10 Truck 352 288 9 5 0 1
Elephant 623 456 60 136 20 19 Turtle 450 311 35 311 70 69
Face 269 337 8 4 0 1 Watch 507 160 35 31 0 1
Euler Characteristic Computation by Means of a Chain Code
2454012-21

stated in Eqs. (10) and (18). In addition, the 2and 2^symbols are thoroughly
characterized. Moreover, it can be appreciated that the number of potholes and
bumps present in the contour of a binary object represented by N2^and N2,
respectively, are even or odd numbers.
4.2. Simpli¯cation of a binary shape
With all the mathematical analysis developed and the results of the tests performed on
the set of images shown in Fig. 13, it is possible to appreciate that each 2or 2^symbol
is located at speci¯c break points that allow characterizing these curves and making the
shape of the object a polygon by joining these points with straight lines. Figure 14(a)
show an example of two binary objects with their symbols 2and 2^identi¯ed clockwise
for the external contour and counterclockwise for the hole contour.
The polygonal approximation resulting from joining these points is also shown in
Fig. 14(b). These vertices become dominant points of the object, which when recon-
structed from them, such approximation does not lose abundant information of its
original shape and faithfully preserves its topology. On the contrary, it results in a good
Fig. 13. Images used to test the proposed method, taken from the MPEG-7 dataset.
35
E. I. G
omez-G
omez & H. S
anchez-Cruz
2454012-22

representation of the original object, thus signi¯cantly reducing the memory storage of
the image (see Table 2) and improving the information handling of the original shape.
In Table 2, a comparison of the bit size of the 3OT chain codes in Fig. 14 is shown,
using three lossless compression entropic compressors, the Context Mixing (CoMi;
see Ref. 31), the Arithmetic Code and Hu®man (see Ref. 30). It is observed that when
applying Hu®man algorithm there is a considerable reduction, more than 97%, of the
image storage when approximating the object using the symbols 2and 2^.
(a) (b)
Fig. 14. Polygonal approximation: (a) original object described in terms of its 2and 2^symbols in
clockwise direction for the external contour and counterclockwise for the hole contour; (b) polygonal
approximation resulting from the dominant points characterized by the symbol 2 of the 3OT chain code.
Table 2. Size in bits of compressed chain with Context Mixing com-
pression, Arithmetic compression and Hu®man compression.
Image 3OT CoMi 3OT Arithmetic 3OT Hu®man
Clover leaf 1,280 1,974 11,602
Clover leaf polygon 984 407 402
Fish 1,176 2,464 8,680
Fish polygon 680 173 167
Euler Characteristic Computation by Means of a Chain Code
2454012-23

The detection of dominant points in the contour of a binary object is a feature
that naturally stands out in the 3OT chain code, due to its composition, since a single
symbol can be placed at the beginning and end points of the here de¯ned staircases
and this gives us a polygonal approximation of the object.
4.3. Classi¯cation of cells in hippocampus region
Neurodegenerative disorders, such as Alzheimer's, Parkinson's and Huntington's
disease, stroke and amyotrophic lateral sclerosis, involve the progressive death of
neurons in di®erent regions of the nervous system. These conditions cause memory
impairment, involuntary and jerky movements, as well as muscle paralysis, among
other cognitive disorders that usually worsen over time and cannot be cured. One
way to assess the damage caused by such disorders is the counting of neuronal cells
using arti¯cial intelligence techniques as in the case of Ref. 45, where authors use
deep learning for counting cells in the hippocampus, in which highly dense cell
population with fuzzy boundaries and low image quality are present. We believe that
classifying the cells present in these samples can provide specialists with extra in-
formation in the diagnosis of these disorders. Thus, making use of Ref. 45 images and
the Parity Theorem 1proposed here, we obtain both N2and N2^in each cell and
classify them.
To perform this task, we use a sample image of size 512 512 pixels and seg-
mented by authors in Ref. 45. Figure 15(a) shows cells from the hippocampus region,
a highly dense area with wide di®erences in morphology and color, whereas image in
Fig. 15(b) corresponds to the segmented image showing the regions of cells. We
identi¯ed each cell numerically to visually classify them, as shown in Fig. 16. Since
(a) (b)
Fig. 15. Ground Truth Enhancement Process for one image with a highly cell population and blurry
boundaries: (a) Original images and (b) manually annotated ground truth images.
E. I. G
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some cells at the edges of the image are truncated, we do not take them into account
for classi¯cation.
As a ¯rst step, the cells were encoded with the 3OT chain code to obtain the N2^
and N2symbols presents in the contour of each of the cells using the Parity
Theorem 1. Table 3shows the results of characterizing the curvatures of the contours
of each cell belonging to Fig. 16. It is observed that these quantities do not exceed six
2and ten 2^symbols.
So, with the help of one of the geometries described in Sec. 2, in addition to taking
into account that there is a relationship between N2^and N2resulting from the
encoding of the cell contour, and ¯nally, considering the relationship between the
contour perimeter and the contact perimeter described in Sec. 3.2, we propose to give
a simple classi¯cation to our set of cells. For this purpose, we de¯ne a classi¯cation,
de¯ning three di®erent groups that we call nibbled,elongated and circular, respec-
tively, and that we explain as following.
Using the bump geometry it is possible to know how nibbled a cell is by observing
that, in our sample, the di®erence between N2^and N2is four, and also that N2^is
greater than N2(see Table 3). Thus, to obtain a normalized nibbled measure we
Fig. 16. Cell numbering.
Euler Characteristic Computation by Means of a Chain Code
2454012-25

used the ratio given in the following equation:
r¼N2
N2^;ð28Þ
where ris zero when N2is zero, however it tends to one when N2^and N2are very
large (remember that the di®erence is always four), therefore rtakes real values in
the interval: ½0;1Þ. Now, we de¯ne a threshold, greater than or equal to 0.4825, to
identify nibbled shapes in our sample (see Fig. 17(c)).
On the other hand, we were able to identify other two groups: circular and
elongated. The previous ratio is not enough to identify the mentioned two groups.
Such is the case of cells 5 and 10 (see Fig. 16), both with the same ratio of 0 (see
Table 3). It would be thought that the frequency of occurrence of the symbols 0 and 1
of the 3OT chain code of each cell could solve this problem, but this is not the case,
on the contrary, cells like 5 and 34, both have a frequency of zeros equal to 0.45 and a
Table 3. Number of two-concave and two-convex and their nibbled cell ratio measure.
Cell N2^N2rCell N2^N2rCell N2^N2r
Cell 1 4 0 0 Cell 19 5 1 0.2 Cell 37 6 2 0.3333
Cell 2 5 1 0.2 Cell 20 4 0 0 Cell 38 8 4 0.5
Cell 3 7 3 0.4285 Cell 21 4 0 0 Cell 39 8 4 0.5
Cell 4 5 1 0.2 Cell 22 5 1 0.2 Cell 40 4 0 0
Cell 5 4 0 0 Cell 23 5 1 0.2 Cell 41 6 2 0.3333
Cell 6 5 1 0.2 Cell 24 6 2 0.3333 Cell 42 7 3 0.4285
Cell 7 5 1 0.2 Cell 25 4 0 0 Cell 43 7 3 0.4285
Cell 8 5 1 0.2 Cell 26 5 1 0.2 Cell 44 8 4 0.5
Cell 9 7 3 0.4285 Cell 27 10 6 0.6 Cell 45 7 3 0.4285
Cell 10 6 2 0.3333 Cell 28 4 0 0 Cell 46 4 0 0
Cell 11 6 2 0.3333 Cell 29 4 0 0 Cell 47 5 1 0.2
Cell 12 9 5 0.5556 Cell 30 4 0 0 Cell 48 9 5 0.5556
Cell 13 7 3 0.4285 Cell 31 5 1 0.2 Cell 49 5 1 0.2
Cell 14 7 3 0.4285 Cell 32 4 0 0 Cell 50 5 1 0.2
Cell 15 6 2 0.3333 Cell 33 6 2 0.3333 Cell 51 7 3 0.4285
Cell 16 6 2 0.3333 Cell 34 5 1 0.2 Cell 52 7 3 0.4285
Cell 17 7 3 0.4285 Cell 35 6 2 0.3333 Cell 53 6 2 0.3333
Cell 18 6 2 0.3333 Cell 36 7 3 0.4285 Cell 54 8 4 0.5
(a) (b) (c)
Fig. 17. Cell sorting: (a) nibbled; (b) elongated and (c) circular.
E. I. G
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frequency of ones equal to 0.52, but each one with a di®erent shape (see Fig. 16),
which makes their classi¯cation di±cult.
To distinguish a circular cell shape from an elongated one, we focus our attention
on the distribution of the pixels that compose it and how they relate to each other,
i.e. how much the pixels touch each other.
Then, to perform such a task, we work with the total number of edges, n1, of the
object, and take into account local properties such as the contour perimeter and
contact perimeter. So, we consider Eq. (25) of Sec. 3.2, where n1is equal to the sum of
the contour perimeter plus the contact perimeter of the object, and ¯nally, the area of
the cell, i.e. the total number of pixels (n2), was also considered in this study.
The contact perimeter described by Eq. (24) implicitly tells us how compact the
object is, i.e. how many contact edges are there in relation to the contour perimeter.
The more contact edges a shape has, the more compact it is. Therefore, this property
can be measured using the de¯nition of discrete compactness given by Ref. 13.
Hence, the measure of discrete compactness CDfor a binary object composed of n2
pixels, is given by
CD¼n2P=4
n2ffiffiffiffiffi
n2
p;ð29Þ
where CDis a continuous value between 0 and 1. CDin Eq. (29) is normalized to a
square composed of n2pixels, where n2is square, also. However for circle shapes CD
takes values close to 1. For two or more disconnected pixels, CDtakes the value zero.
In our case, the discrete compactness of each of the cell shapes is obtained and it is
observed that the minimum and maximum compactness measures of the set of cells
shapes is 0.9608 and 0.9960 (see Table 4), respectively. Of course, in the universe of
shapes composed with the number of pixels of our sample, we have a small set with
high compactness. However, from the second and third digits of this measurement we
can detect certain appreciable di®erences in the shapes. Consequently, to determine
whether a cell is circular or elongated using this measure, we de¯ne a threshold where
Table 4. Number of two-concave and two-convex, measure of nibbled cell ratio and the discrete compactness
of cells not classi¯ed as nibbled.
Cell N2^N2rCDCell N2^N2rCDCell N2^N2rCD
Cell 1 4 0 0 0.9757 Cell 19 5 1 0.2 0.9835 Cell 32 4 0 0 0.9688
Cell 2 5 1 0.2 0.9779 Cell 20 4 0 0 0.9926 Cell 33 6 2 0.3333 0.9835
Cell 4 5 1 0.2 0.9925 Cell 21 4 0 0 0.9904 Cell 34 5 1 0.2 0.9821
Cell 5 4 0 0 0.9846 Cell 22 5 1 0.2 0.9762 Cell 35 6 2 0.3333 0.9908
Cell 6 5 1 0.2 0.9936 Cell 23 5 1 0.2 0.9881 Cell 37 6 2 0.3333 0.9929
Cell 7 5 1 0.2 0.9943 Cell 24 6 2 0.3333 0.9870 Cell 40 4 0 0 0.9608
Cell 8 5 1 0.2 0.9960 Cell 25 4 0 0 0.9916 Cell 41 6 2 0.3333 0.9875
Cell 10 6 2 0.3333 0.9863 Cell 26 5 1 0.2 0.9938 Cell 46 4 0 0 0.9864
Cell 11 6 2 0.3333 0.9929 Cell 28 4 0 0 0.9900 Cell 47 5 1 0.2 0.9882
Cell 15 6 2 0.3333 0.9869 Cell 29 4 0 0 0.9610 Cell 49 5 1 0.2 0.9946
Cell 16 6 2 0.3333 0.9924 Cell 30 4 0 0 0.9715 Cell 50 5 1 0.2 0.9958
Cell 18 6 2 0.3333 0.9912 Cell 31 5 1 0.2 0.9618 Cell 53 6 2 0.3333 0.9892
Euler Characteristic Computation by Means of a Chain Code
2454012-27

cell shapes with compactness smaller than 0.9917 belong to the class of elongated
shapes (see Fig. 17(b)), while cell shapes with compactness greater than or equal to
0.9917 (see Table 4) belong to the class of cells with circular shape (see Fig. 17(a)).
Of course, this classi¯cation can be subjective, i.e. it depends on the sample
shapes being worked with and therefore, the specialist can adjust the classi¯cation
thresholds described above accordingly.
4.4. Comparison of the proposed method with others
With the intention of completing the tests performed in the previous section, in which
the robustness of our method is demonstrated against a set of images with varied
characteristics, in this section we present a qualitative and quantitative analysis that
aims to compare our method in a fair way with other proposals based on chain code and
the perimeter of a binary object, highlighting the characteristics of each method.
4.4.1. Qualitative comparison
To compare our proposal with the methods based on chain codes, we selected the
works proposed by Refs. 43 and 15. In Ref. 43, authors provide two new equations for
the computation of the Euler characteristic of a binary image composed of square
cells and hexagonal cells. In addition, they provide a unifying method for both
equations. These formulas are duly demonstrated and numerically validated with
simple examples and a set of binary images. The works of Refs. 15 and 43 were
selected because interesting properties of binary objects are described using the VCC,
such as the analysis of complementary chains, the calculation of connected regions
with holes, and the detection of convex and concave shapes. The di®erences between
these methods comparing with ours are described in Table 5. For example, we
compare the number of symbols used to obtain the Euler characteristic, the type of
tessellation that covers the image and with which a chain code can be obtained, the
invariance to certain geometric transformations, the type of connectivity that each
Table 5. Comparison of our proposal and the VCC-based methods.
Features Ref. 43 Ref. 15 Proposed
Symbols used for the calculation of the Euler characteristic 2 2 1
Square tessellation XX X
Hexagonal tessellation X
Invariant to scale XX X
Invariant to rotation XX X
Invariant to mirror XX X
Four connectivity XX X
Eight connectivity  X
Dependence on the symbol above  X
Independence of starting point XX X
Complementary chains detection XX
Global concavities/convexities  X
Geometries in the contour  X
E. I. G
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method can handle, and whether or not the method is dependent on any symbol or its
starting point, whether it detects convexities or concavities in the object contour, or
whether it ¯nds complementary chains or any particular geometry.
It is interesting to compare our method with proposals that have been developed
with respect to the perimeter and contact perimeter of a binary object since they are
analogous to the 3OT chain code that we are using here, i.e. they work with the edge
of a pixel and ¯nd characteristics and relationships between pixels that allow de-
veloping an equation for the calculation of the topological descriptor in question.
Such is the case of Ref. 14, which was the ¯rst proposal developed in these terms and
was de¯ned for 2D and 3D unit shapes connected laterally and in turn, for di®erent
types of cells. Later in Ref. 42, the proposal presented in Ref. 14 was further extended
to shapes of any thickness and also proposed a structuring element called tetrapixel
that contributes to the calculation of the Euler characteristic.
4.4.2. Quantitative comparison
From Freeman chain codes of eight and four directions, called F8 and F4, respec-
tively (Ref. 22), to perform this comparison, we employ the set of images in Fig. 13
described now in terms of the VCC with the intention of performing a direct com-
parison with another chain code. Since there are transition matrices to go from one
chain code to another (see Ref. 38), we obtain the complexity of our method as
follows. First, we ¯nd the simplest code to implement, F8, for this we visit the image,
pixel by pixel, to follow the contour of the binary object contained in it, which takes a
time of OðMNÞ, where Mand Nare the number of rows and columns of the image,
respectively. Later, we convert F8 to F4 through the matrix (28),
0 010 01 0121 03
10 1 121 12 3
10 1 121 12 1232 
021 2 232 23 
21 2 232 23 2303
30 132 3 303
30 2010 32 3 303
0 010 01 203
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
ð30Þ
which leads us to read each of the symbols of F8 and compare with the matrix. This
takes OðnÞtime, where nis the length of the string, which is equivalent to the
perimeter of the binary object, which is less than MN. Finally, we convert F4 to
VCC and F4 to 3OT through the matrix (29) and (30), respectively,
012
201
201
120
0
B
B
@
1
C
C
A
;ð31Þ
Euler Characteristic Computation by Means of a Chain Code
2454012-29

12
12
21
21
0
B
B
@
1
C
C
A
ð32Þ
which also takes OðnÞtime in both cases. So, the complexity of both methods is
OðMNÞþ3OðnÞ¼OðMNÞ. Some additional operations needed to ¯nd the
Euler characteristic, in Refs. 17 and 47, the calculation of the Euler characteristic
consists of counting the number of symbols 1 and 2 present in the VCC string of the
contour of a binary object. The complexity of this task is to OðnÞ. Compared to our
method it is quite similar, since the complexity of the proposed Parity Theorem 1is
also OðnÞ. The execution time for the di®erent methods is quite similar, they are in
the order of 1 104s. However, our method employs only one symbol for the
computation of the Euler characteristic and also the number of convexities ðN2^Þ
and concavities ðN2Þis smaller than the number of symbols 1 and 2 of the VCC used
in the computation of the Euler characteristic.
4.5. Advantages of the proposed approach
We have found an alternative way to calculate the Euler characteristic, a di®erent
method that employs a chain code with outstanding features for image compression,
the 3OT. The way this chain encodes the contour of a binary object allows us to
identify four geometries: plains, staircase, potholes and bumps, which are relevant for
pattern recognition, because they identify complete parts of the object, without
jumping from one symbol to another in sections such as staircases (see Fig. 18(a)
(a) (b)
Fig. 18. Comparison of di®erences between the chain codes: (a) 3OT and (b) VCC.
E. I. G
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omez & H. S
anchez-Cruz
2454012-30

yellow color), where the VCC uses two symbols to represent them (see Fig. 18(b)
yellow color), or identify convexities and concavities in the shape of the object in
general with bumps and potholes, identifying these two types of curvatures from a
3OT code symbol using dynamic rules that distinguish a two-concave symbol from a
two-convex symbol.
Unlike the VCC that identi¯es a convexity from a pixel (see Fig. 18(b) red color)
and a concavity from the adjacency the edges of three pixels (see Fig. 18(b) gray
color) to count them and then obtain the Euler characteristic, this method has the
ability to join two concavities or two contiguous convexities regardless of whether or
not there is a plain between them, to form a pothole or a bump, respectively, and
thus count them to obtain the Euler characteristic.
Interested reader can reproduce experiments with our proposals using our source
code: https://github.com/EliGGomez/Euler-Characteristic-3OT.git.
5. Conclusion and Future Work
In this paper, a new equation is proposed for the calculation of the Euler charac-
teristic for four or eight-connected 2D binary objects, by employing a single symbol
of the chain code 3OT. This is being achieved by means of a Parity Theorem that
discriminates this symbol as a concavity or a convexity
Four geometries are also proposed, plain, staircase, bump and pothole that extend
the local characteristics of a binary object proposed in the literature to global
characteristics of the object shape.
That is, with our proposal, it is no longer necessary to work at the pixel level as in
previous proposals using patterns, perimeters, component labeling or skeletons to
describe it topologically, but with complete pieces of the object's contour, in this case
staircase, bumps, pothole and plain contours, which allow us to recognize it unam-
biguously using fewer features and less data, speci¯cally with a symbol. In addition,
the proposed Parity Theorem 1distinguishes two types of curvatures, which are
representative and in turn preserve the most signi¯cant features of the shape of a 2D
object that in computer vision are named dominant points and are used to digitize an
image and reconstruct with less information the binary objects. With this, we can not
only describe the object topologically, but also geometrically.
To prove our method, we have used an arbitrary set of binary objects with a
variable number of holes and the results obtained with the simpli¯ed formulas are
faithful to those obtained with the Euler equation for 2D images.
As future work, we are aiming to extend the concepts of staircases, bumps and
potholes to calculate the Euler characteristic to 3D objects composed by voxels.
Acknowledgment
Second author thanks Universidad Autónoma de Aguascalientes for its support
under Grant PII24-4.
Euler Characteristic Computation by Means of a Chain Code
2454012-31

Appendix A
Theorem A.1. The Euler characteristic of a binary object considering both local
properties and neighborhoods is given by
E¼n0n1þn2þn:ðA:1Þ
Proof. Let's do the proof by induction on the number of pixels, n2.
Basis step:n2¼1. In this case, one pixel has four vertices and four edges, i.e.
n0¼4, n1¼4 and n¼0, respectively. So, E¼1, and the theorem is valid.
Induction hypothesis. Suppose the theorem is valid for n2¼n.
Let us demonstrate that it is valid for n0
2¼nþ1.
The nþ1 pixel can be connected in one of the di®erent ways shown in Fig. A.1.
Let us calculate, then, the increase of E,E.
(a) Since n0
0¼3, n0
1¼4, n0
2¼nþ1 and n¼1)n0¼3, n1¼4 and
n2¼1, n¼1.
)34þ1þ1¼1)E¼1. So, the theorem is valid for nþ1.
(b) Then n0
0¼2, n0
1¼4, n0
2¼nþ1andn¼2)n0¼2, n1¼4 and
n2¼1, n¼2.
)24þ1þ2¼1)E¼1. So, the theorem is valid for nþ1.
(c) Now n0
0¼1, n0
1¼4, n0
2¼nþ1andn¼3)n0¼1, n1¼4 and
n2¼1, n¼3.
)14þ1þ3¼1)E¼1. So, the theorem is valid for nþ1.
(d) Finally n0
0¼0, n0
1¼4, n0
2¼nþ1andn¼4)n0¼0, n1¼4 and
n2¼1, n¼4.
)04þ1þ4¼1)E¼1. So, the theorem is valid for nþ1.
Theorem A.2. The number of 22arrays,called tetrapixeles,i.e.22turned on
neighbors,is given by
Nt¼n02n1þ4n2þn:ðA:2Þ
Proof. Basis step:n2¼1. In this case, one pixel has four vertices and four edges, i.e.
n0¼4andn1¼4, respectively. So, Nt¼0, and the theorem is valid.
(a) (b) (c) (d)
Fig. A.1. Di®erent possibilities to connect an extra pixel to a n2pixel object: (a) by one vertex, (b) by two
vertices, (c) by three vertices and (d) by four vertices.
E. I. G
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omez & H. S
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2454012-32

Induction hypothesis. Suppose the theorem is valid for n2¼n.
Let us demonstrate that it is valid for n0
2¼nþ1.
The nþ1 pixel can be connected in one of the di®erent ways shown in Fig. A.2.
Let us calculate, then, the increase of Nt,Nt.
(a) Since n0
0¼1, n0
1¼2, n0
2¼nþ1)n0¼1, n1¼2 and n2¼1, n¼0.
)12ð2Þþ4ð1Þþ0¼1)Nt¼1. So, the theorem is valid for nþ1.
(b) In this case, n0¼2, n1¼3, n2¼1andn¼0.
)22ð3Þþ4ð1Þþ0¼0)Nt¼0. So, the theorem is valid for nþ1.
(c) The proof is similar to the previous case.
(d) Here, n0¼3, n1¼4, n2¼1 and n¼1.
)32ð4Þþ4ð1Þþ1¼0)Nt¼0. So, the theorem is valid for nþ1.
ORCID
Elisa I. Gómez-Gómez https://orcid.org/0009-0001-8684-8752
Hermilo S
anchez-Cruz https://orcid.org/0000-0001-9081-6449
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Elisa I. G
omez-G
omez
obtained her M.Sc. in
Science, with Options in
Computer Science,
Applied Mathematics in
2023 from the Uni-
versidad Autónoma de
Aguascalientes (UAA).
She is currently Ph.D.
candidate, also at UAA.
She has worked as Aca-
demic Technician in projects related to image
compression, as well as in the search of descrip-
tors for 2D and 3D objects. She is interested in
solving problems in computer vision, pattern
recognition, image analysis, including aspects of
2D and 3D object topology and neural networks.
Hermilo S
anchez-Cruz
received his Ph.D. in
Computer Science in 2002
and his B.Sc. in Physics in
1995, both from the Uni-
versidad Nacional Autón-
oma de M
exico (UNAM).
Also, he was Assistant
Researcher with the
Instituto de Investiga-
ciones en Matem
aticas
Aplicadas y en Sistemas at the UNAM, where he
took part in projects about biomedical images.
Currently, he is Full-time Professor with the
Universidad Autónoma de Aguascalientes in
M
exico (UAA), where he teaches graduate and
undergraduate courses in pattern recognition
and image processing. He is Member of the
National Research System of M
exico since 2005.
His areas of interest are pattern recognition,
image compression, bidimensional and 3D image
recognition.
E. I. G
omez-G
omez & H. S
anchez-Cruz
2454012-36