Topology-Preserving Reductions on (18,12) Pictures of the

Face-Centered Cubic Grid

G´abor Karai

, P´eter Kardos

and K´alm´an Pal´agyi

Department of Image Processing and Computer Graphics, University of Szeged, Szeged, Hungary

Keywords:

Shape Representation, Feature Selection and Extraction, FCC Grid, Topology Preservation.

Abstract:

Reductions transform binary pictures only by changing some black points to white ones. Topology preserva-

tion is a major concern of thinning algorithms that are composed of reductions. For (18, 12) binary pictures

on the 3D face-centered cubic (FCC) grid, we propose four sufﬁcient conditions for topology-preserving par-

allel reductions that can change a set of black points simultaneously. The ﬁrst two conditions examine some

conﬁgurations of changed points, and they provide methods of verifying that formerly constructed parallel

reductions preserve the topology. The further two conditions focus on individual points, directly provide dele-

tion rules of topology-preserving parallel reductions, and make us possible to establish topologically correct

parallel thinning algorithms.

1 INTRODUCTION

It is the common practice that 3D digital pictures are

sampled on the cubic grid Z

, since it is the only regu-

lar grid in 3D, it has a fairly simple structure, and digi-

tal pictures on the cubic grid can be naturally stored in

usual 3D arrays. Among non-standard grids our atten-

tion is focused on the face-centered cubic (FCC) grid

(Kong and Rosenfeld, 1989). The points (x, y, z) ∈Z

such that x + y+ z is even are the grid points of the

FCC grid denoted by F.

Herman touched upon some disadvantages of the

cubic grid (Herman, 1998), Gau and Kong reported

three advantages of the FCC grid over the cubic

grid (Gau and Kong, 1999), and Edelsbrunner et al.

showed that the FCC grid provides the densest sphere

packing (Edelsbrunner et al., 2015). That is why the

importance of the FCC grid shows an upward ten-

dency (

Comi´c and Magillo, 2020;

Comi´c and Nagy,

2016; Gastineau and Togni, 2021; Koshti et al., 2018;

R´acz and Cs´ebfalvi, 2018; Strand and Stelldinger,

2007).

A binary digital picture (or picture, for short) on

a discrete grid is composed of black or white points

(Kong and Rosenfeld, 1989). Reduction operators

transform a picture by changing some black points

to white ones that is referred to as deletion, while all

https://orcid.org/0000-0001-9609-8628

https://orcid.org/0000-0001-8857-4102

https://orcid.org/0000-0002-3274-7315

white points remain unchanged (Hall, 1996). Parallel

reductions can delete a set of points simultaneously,

while sequential reductions traverse the black points

of a picture, and focus on the actually visited single

point for possible deletion (Hall, 1996).

Thinning algorithms iteratively apply reductions

(Saha et al., 2016), and a crucial issue in thinning

is to ensure topology preservation (Kong and Rosen-

feld, 1989). The problems of verifying that ex-

isting 3D parallel thinning algorithms always pre-

serve the topology (Kong, 1995) and how to con-

struct such topologically correct algorithms (Pal´agyi

et al., 2012) have been solved for pictures on the

traditional cubic grid. Since it cannot be abso-

lutely said in the case of the FCC grid, this paper

establishes four sufﬁcient conditions for topology-

preserving parallel reductions acting on this uncon-

ventional 3D grid. The ﬁrst two conditions focus

on some conﬁgurations of deleted points. Thus they

state conﬁguration-based results, and they are suit-

able for verifying that a formerly constructed 3D re-

duction is topology-preserving for all possible pic-

tures on the FCC grid. Since the remaining two con-

ditions examine the deletability of individual points,

they are said to be point-based sufﬁcient conditions

for topology-preserving reductions on the FCC grid.

They directly provide deletion rules of parallel reduc-

tions, and make us possible to generate various topo-

logically correct parallel thinning algorithms.

The rest of this paper is organizedas follows: Sec-

tion 2 gives an overview of the basic notions and re-

254

Karai, G., Kardos, P. and Palágyi, K.

Topology-Preserving Reductions on (18,12) Pictures of the Face-Centered Cubic Grid.

DOI: 10.5220/0011633500003411

In Proceedings of the 12th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2023), pages 254-261

ISBN: 978-989-758-626-2; ISSN: 2184-4313

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

sults. Then, in Section 3 the two conﬁguration-based

sufﬁcient condition for topology-preserving parallel

reductions on the FCC are proposed. Section 4

presents our point-based sufﬁcient conditions, and we

generate directly two topology-preserving parallel re-

ductions in Section 5. Finally, we round off this work

with some concluding remarks.

2 BASIC NOTIONS AND

EXISTING RESULTS

Next, we deﬁne the key concepts of digital topology

as reviewed in (Kong and Rosenfeld, 1989), and recall

the previously stated results that we need later on.

The Voronoi neighborhood of a point p ∈ F is

the set of all points in the 3D Euclidean space that

are at least as close to p as to any other point in

the FCC grid (Kong and Rosenfeld, 1989). It is the

rhombic dodecahedron with twelve faces centered on

p, and it is called an FCC-voxel (voxel, for short).

As it is illustrated by Fig. 1, there are exactly two

Voronoi adjacency relations on the FCC grid: 12-

adjacency and 18-adjacency. The set N

(p) contains

the twelve grid points that are at a distance of

√

2 from

p. The set N

(p) is formed of the six grid points

that are at a distance of 2 from p together with the

twelve grid points in N

(p). Elements of N

(p) and

(p) are 12-adjacent and 18-adjacent to p, respec-

tively. Note that each voxel meets eighteen others, see

Fig. 1. The voxel associated with p shares a face with

the twelve voxels corresponding to the grid points in

(p), and it shares just a vertex with the six voxels

of N

(p) \N

(p).

Figure 1: The studied adjacency relations on F. The twelve

points marked ‘’ form the set N

(p), and N

(p) con-

tains six more points marked ‘♦’ (top). (Note that un-

marked elements of Z

are not grid points in F.) The voxel-

representations of N

(p) (bottom left) and N

(p) \N

(p)

(bottom right), where each voxel is a rhombic dodecahe-

dron.

An (m, n) picture on the FCC grid is a quadru-

ple (F, m, n, B) (Kong and Rosenfeld, 1989), where

B ⊆ F denotes the set of black points; each point

in F \B is said to be a white point; adjacency rela-

tions m and n are assigned to B and F \B, respec-

tively. In their seminal work, Gau and Kong exam-

ined three types of pictures on the FCC grid: (m, n) =

(18, 12), (12, 18) and (12, 12) (Gau and Kong, 1999).

Since our attention is focused on the (18, 12) pictures,

in the rest of this paper, B denotes the set of black

points in the picture (F, 18, 12, B).

Since both of the studied relations are symmetric,

their transitive closures form equivalence relations,

and their equivalence classes are called components.

A black component or an object is an 18-component

of B, while a white component is a 12-component of

the set of white points F \B.

A point p ∈ B is a border point if N

(p) ∩(F \

B) 6=

0 (i.e., p is 12-adjacent to at least one white

point), it is an interior point if N

(p) ⊂ B (i.e., is

not a border point), and it is an isolated point if

(p) ∩B =

0 (i.e., p forms a singleton object).

Gau and Kong introduced a further concept (Gau

and Kong, 1999): A set of mutually 18-adjacent

points is called a small set. It can be readily seen

that a small set may consist of at most six points, see

Fig. 2. Note that a small set is an 18-component, but

it does not need to be a 12-component.

Figure 2: The three possible maximal small sets. All small

sets are (nonempty) subsets of them.

A crucial issue in thinning algorithms, composed

of reductions, is to ensure topology preservation

(Kong and Rosenfeld, 1989; Kong, 1995). A 2D re-

duction is topology-preserving if and only if any ob-

ject in the input picture contains exactly one object in

the output picture, and any white component in the

output picture contains exactly one white component

in the input picture. There is an additional concept

called hole (or tunnel) in 3D pictures. A hole (which

donuts have) is formed from white points, but it is not

a white component (Kong and Rosenfeld, 1989). To

preserve topology, a 3D reduction must not create nor

eliminate any hole.

There is a key concept in digital topology called

a simple point. A simple point in a picture is a black

Topology-Preserving Reductions on (18,12) Pictures of the Face-Centered Cubic Grid

255

point whose deletion is a topology-preserving reduc-

tion (Kong and Rosenfeld, 1989). Gau and Kong

gave the following characterization of simple points

in (18, 12) pictures:

Theorem 1. (Gau and Kong, 1999) A point p ∈ B is

simple for B if and only if the following conditions

hold:

1. N

(p) ∩B contains exactly one 18-component.

2. N

(p) \B contains exactly one 12-component.

Theorem 1 implies that only non-isolated border

points may be simple, and simple points can be lo-

cally characterized (i.e., the simpleness of a point p

can be decided by examining the points in N

(p)).

Figure 3 givesfour illustrative examples of simple and

non-simple points.

Figure 3: Examples of simple and non-simple points in

(18, 12) pictures. The positions denoted by ‘•’ and ‘◦’ refer

to black and white points, respectively. Black point p is sim-

ple only in the top left conﬁguration. In the top right case, p

is an isolated black point, while in the bottom left example,

(p) ∩B contains two 18-components, hence both cases

violate condition 1 of Theorem 1. In the bottom right ﬁgure,

there are two 12-components in N

(p) \B, thus condition

2 of Theorem 1 does not hold.

It is obvious that a sequential reduction (or a thin-

ning algorithm composed of sequential reductions)

preserves the topology if and only if it deletes only

simple points. Unlike the sequential case, parallel

reductions can delete a set of points simultaneously.

Thus we need to consider what is meant by topology

preservation when more than one point is deleted at a

time.

We are to deﬁne the concepts of a simple set, a

simple sequence, and a minimal non-simple set.

Deﬁnition 1. (Kong, 1995) Let P be an arbitrary pic-

ture. A set of k black points Q is a simple set in P if it

is possible to arrange the elements of Q in a sequence

, . . . , q

i such that q

is simple in P and each q

simple after the set of points {q

, . . . , q

i−1

} is deleted

(i = 2, . . . , k). Such a sequence is called a simple se-

quence. (And let the empty set be simple.)

Deﬁnition 2. (Ronse, 1988) A set of black points is

a minimal non-simple (MNS) set in an arbitrary pic-

ture if it is not simple, but any of its proper subsets is

simple.

Figure 4 presents examples of simple, non-simple,

and MNS sets in an (18, 12) picture.

Figure 4: Examples of simple and non-simple sets. The set

of black points {a,b, c, d} is simple since the 16 sequences

(of the possible 24 ones) ha, b, c,di, ha, b, d, ci, ha, c, b, di,

ha, c, d, bi, ha, d, b, ci, ha, d, c, bi, hb, a, c, di, hb, a, d, ci,

hb, d, a, ci, hc, a, b, di, hc, a, d, bi, hc, d, a, bi, hd, a, b, ci,

hd, a, c, bi, hd, b, a, ci, hd, c, a, bi are simple. The set {b, c}

is minimal non-simple, since both sequences hb, ci and

hc, bi are non-simple and both proper subsets {b} and {c}

of {b, c} are simple. The set {b, c, d} is non-simple but not

minimal non-simple, since {b,c} is its proper non-simple

subset. Note that points a, b, d, e, and f are all 18-adjacent

to point c.

We state the following proposition that is a

straightforward consequence of Deﬁnition 1:

Proposition 1. Let Q ⊂ B be a simple set for B. If

p ∈(B\Q) is a simple point for B\Q, Q∪{p}is also

a simple set for B.

Here, we recall a general lemma stated by Kardos

and Pal´agyi:

Lemma 1. (Kardos and Pal´agyi, 2015) Let p and q

be two black simple points in an arbitrary picture. If

p remains simple after the deletion of q, q remains

simple after the deletion of p.

In other words, the simpleness of a set of two sim-

ple points can be decided by examining just one se-

quence of its elements.

The following theorem gives a universal sufﬁcient

condition for topology-preservingparallel reductions:

Theorem 2. (Ronse, 1988) A reduction preserves the

topology for an arbitrary picture if it does not delete

any MNS set.

ICPRAM 2023 - 12th International Conference on Pattern Recognition Applications and Methods

256

Gau and Kong stated the following speciﬁc prop-

erty of MNS sets for (18, 12) pictures:

Proposition 2. (Gau and Kong, 1999) If Q ⊆ B is an

MNS set, Q is a small set.

They also identiﬁed the set of points that can be

MNS sets:

Theorem 3. (Gau and Kong, 1999) Let Q ⊆ B be a

nonempty small set of black points. Then Q satisﬁes

exactly one of the following conditions:

1. Q is a subset of three mutually 12-adjacent points.

2. Q is a set of four mutually 12-adjacent points.

3. The points in Q are not mutually 12-adjacent

(i.e., there are two points p, q ∈ Q such that q 6∈

(p)).

If Q satisﬁes condition 1 then Q may be an MNS set

without being an object. If condition 2 or condition

3 holds then Q is an MNS set if and only if Q is an

object.

3 CONFIGURATION-BASED

SUFFICIENT CONDITIONS

As a consequence of Theorem 3, we derived the fol-

lowing sufﬁcient condition for topology-preserving

parallel reductions:

Theorem 4. A parallel reduction R is topology-

preserving for B ⊂ F if the following conditions hold:

1. Any set of at most three mutually 12-adjacent

points Q ⊆ B deleted by R is simple.

2. R does not delete completely any object of B com-

posed of four mutually 12-adjacent points (see the

last two small objects in Fig. 2).

3. R does not delete completely any small object in

which the points are not mutually 12-adjacent(see

Fig. 5).

Proof. To prove this theorem, we must show that R

does not delete any MNS set (as it is required by The-

orem 2). Note that all possible MNS sets are charac-

terized by Theorem 3.

It can be readily seen that if condition i (i = 1, 2, 3)

of this theorem holds, R does not delete any MNS set

that is speciﬁed by condition i of Theorem 3.

We can state that Theorem 4 takes some conﬁgu-

rations of at most six points into consideration. Thus

our theorem states a conﬁguration-based sufﬁcient

condition for topology-preserving parallel reductions

(acting on (18, 12) pictures of the FCC grid). Notice

that, by condition 1 of Theorem 4, it is difﬁcult to

verify that an existing parallel reduction (or thinning

Figure 5: Six of the possible 37 small objects in which the

points are not mutually 12-adjacent (see condition 3 of The-

orem 4). The remaining ones are rotated and reﬂected ver-

sions of these six base objects.

algorithm) preserves the topology for all possible pic-

tures. That is why, with the help of Proposition 1 and

Lemma 1 we propose a simpliﬁed version of the very

ﬁrst conﬁguration-based sufﬁcient condition:

Theorem 5. A parallel reduction R is topology-

preserving for B ⊂ F if the following conditions hold:

1. Only simple points for B are deleted by R .

2. If two 12-adjacent points p and q are deleted by

R , p is simple for B\{q}.

3. If three mutually 12-adjacent points p, q, and r

are deleted by R ,

p is simple for B\{q, r}, or

q is simple for B\{p, r}, or

r is simple for B \{p, q}.

4. R does not delete completely any object of B com-

posed of four mutually 12-adjacent points.

5. R does not delete completely any small object in

which the points are not mutually 12-adjacent.

Proof. Since the last two conditions of this theorem

are the same as conditions 2 and 3 of Theorem 4, it is

sufﬁcient to show that the ﬁrst three conditions of this

theorem together imply condition 1 of Theorem 4.

Let us suppose that R satisﬁes all conditions of

this theorem, and it deletes the set of points D ⊂ B.

Let Q ⊆ D be a set of at most three mutually 12-

adjacent black points. Then the following three points

are to be investigated:

• Q = {p}:

By condition 1 of this theorem, point p is simple

for B. Thus the singleton set Q is a simple set.

• Q = {p, q}:

By condition 1 of this theorem, both points p and

q are simple for B. By condition 2 of this theo-

rem, p is simple for B \{ q}. Consequently the

set of two points Q is a simple set. (Note that, by

Lemma 1, q is also simple for B \{p}. That is

why we do not need to distinguish p and q.)

• Q = {p, q, r}:

By condition 1 of this theorem, all the three points

p, q, and r are simple for B. By condition 2 of

Topology-Preserving Reductions on (18,12) Pictures of the Face-Centered Cubic Grid

257

this theorem and Lemma 1, all the three sets of

two points {p, q}, {p, r}, and {q, r} are simple

sets, and all the six sequences hp, qi, hq, pi, hp, ri,

hr, pi, hq, ri, and hr,qi are simple sequences. By

condition 3 of this theorem and Proposition 1, at

least two of the six sequences hp, q, ri, hq, p, ri,

hp, r, qi, hr, p, qi, hq, r, pi, and hr, q, pi are simple.

Thus the set of three points Q is a simple set.

Since any set of at most three mutually 12-adjacent

deleted points is a simple set, condition 1 of Theorem

4 also holds.

4 POINT-BASED SUFFICIENT

CONDITIONS

The conﬁguration-based sufﬁcient conditions stated

in Theorems 4 and 5 are capable of verifying the

topological correctness of existing parallel reduc-

tions, however, they do not serve as a methodology

for designing topology-preservingparallel reductions.

For this reason, here we propose point-based sufﬁ-

cient conditions that directly yield deletion rules of

topology-preserving parallel reductions. The follow-

ing two theorems examine the deletability of individ-

ual points:

Theorem 6. A parallel reduction is topology-

preserving for B if each point p ∈ B deleted by this

reduction satisﬁes the following conditions:

1. Point p is simple for B.

2. For any point q ∈ N

(p) ∩B if q is simple for B

then p is simple for B\{q}.

3. For any two points q ∈ N

(p) ∩ B and r ∈

(p) ∩N

(q) ∩B if q and r are simple for B,

and q is simple for B \{r} then p is simple for

B\{q, r}.

4. Point p is not an element of an object consisting

of four mutually 12-adjacent points.

5. Point p is not an element of a small object in

which the points are not mutually 12-adjacent.

Proof. Let us suppose that a parallel reduction satis-

ﬁes all conditions of this theorem, it deletes the set of

points D ⊂ B, and a black point p is in D. To prove

this theorem, we must show that all conditions of The-

orem 5 hold.

• Let Q ⊆ D be a set of at most three mutually 12-

adjacent black points. Then the following three

points need to be investigated:

– Q = {p}:

By condition 1 of this theorem, point p is sim-

ple for B. Thus condition 1 of Theorem 5 holds.

– Q = {p, q}.

By condition 1 of this theorem, q is a simple

point for B. By condition 2 of this theorem,

p is simple for B \{q}. Thus condition 2 of

Theorem 5 is satisﬁed.

– Q = {p, q, r}.

By condition 1 of this theorem, q and r are both

simple points for B. By condition 2 of this the-

orem, q is simple for B\{r}. By condition 3 of

this theorem, p is simple for B \{q, r}. Hence

condition 3 of Theorem 5 holds.

• By condition 4 of this theorem, none of the el-

ements of an object consisting of four mutually

12-adjacent points may be deleted. Since such an

object cannot be deleted completely, condition 4

of Theorem 5 is satisﬁed.

• By condition 5 of this theorem, none of the ele-

ments of a small object in which the points are not

mutually 12-adjacent may be deleted. Since such

objects cannot be deleted completely, condition 5

of Theorem 5 is also satisﬁed.

Since all the ﬁve conditions of Theorem 5 hold, this

theorem is true.

Conditions of Theorem 6 may be viewed as sym-

metric since elements in the examined sets points are

not distinguished.

Let us focus on the addressing scheme shown in

Fig. 6, which maps every point in F to a triplet of in-

teger coordinates. The lexicographical order relation

‘≺’ between two distinct points p = (p

, p

) and

q = (q

, q

) is deﬁned as follows:

p ≺ q ⇔ (p

< q

) ∨(p

= q

∧p

< q

) ∨

= q

∧p

= q

∧p

< q

)

Figure 6: The considered coordinate system (left) and the

ordering scheme for the FCC grid (right). The elements

of the set of nine points { q | q ∈ N

(p), p ≺ q} are

marked ‘’, and the remaining nine points in the set { r |r ∈

(p), r ≺ p} are marked ‘♦’.

Let Q ⊂ F be a ﬁnite set of points. Point p ∈ Q

is the smallest element of Q if for any q ∈ Q\{p},

p ≺ q.

With the help of the proposed ordering (see

Fig. 6), we state the following asymmetric point-

based condition for topology-preserving parallel re-

ductions:

ICPRAM 2023 - 12th International Conference on Pattern Recognition Applications and Methods

258

Theorem 7. A parallel reduction is topology-

preserving for B if each point p ∈ B deleted by that

reduction satisﬁes the following conditions:

1. Point p is simple for B.

2. For any point q ∈ N

(p) ∩B if q is simple for B

then p is simple for B\{q}, or q ≺ p.

3. For any two points q ∈ N

(p) ∩ B and r ∈

(p) ∩N

(q) ∩B if q and r are simple for B,

and q is simple for B \{r} then p is simple for

B \ {q, r}, or p is not the smallest element of

{p, q, r}.

4. Point p is not the smallest element of an object

consisting of four mutually 12-adjacent points.

5. Point p is not the smallest element of a small

object in which the points are not mutually 12-

adjacent.

Proof. Let us suppose that a parallel reduction satis-

ﬁes all conditions of this theorem, and it deletes the

set of points D ⊂ B. To prove this theorem, we must

show that all the ﬁve conditions of Theorem 5 hold.

• Let Q ⊆ D be a set of at most three mutually 12-

adjacent black points. Then the following three

points need to be investigated:

– Q = {u}:

By condition 1 of this theorem, point u is sim-

ple for B. Thus condition 1 of Theorem 5 holds.

– Q = {t, u}, where t ≺ u.

By condition 1 of this theorem, both points t

and u are simple for B. Since t ≺ u, by condi-

tion 2 of this theorem, t is simple for B \{u}.

Thus condition 2 of Theorem 5 is satisﬁed.

– Q = {s, t, u}, where s ≺t ≺ u.

By condition 1 of this theorem, s, t, and u are

simple points for B. By condition 2 of this the-

orem, t is simple for B \ {u}. Since s is the

smallest element of {s,t,u}, by condition 3 of

this theorem, s is simple for B\{t, u}. Hence

condition 3 of Theorem 5 holds.

• By condition 4 of this theorem, the smallest el-

ement of an object formed of four mutually 12-

adjacent points cannot be deleted. Thus that ob-

ject cannot be deleted completely, and condition 4

of Theorem 5 is satisﬁed.

• By condition 5 of this theorem, the smallest el-

ement of a small object in which the points are

not mutually 12-adjacent cannot be deleted. Thus

that object cannot be deleted completely, and con-

dition 5 of Theorem 5 also holds.

Since all the ﬁve conditions of Theorem 5 are satis-

ﬁed, the proof is completed.

5 TWO GENERATED

TOPOLOGY-PRESERVING

PARALLEL REDUCTIONS

In this section we show that our point-based sufﬁ-

cient conditions (see Theorems 6 and 7) allow us to

construct directly topology-preserving parallel reduc-

tions.

Deﬁnition 3. A black point is deleted by the parallel

reduction S if it satisﬁes all conditions of Theorem 6

(i.e., symmetric point-based condition for topology-

preserving parallel reductions).

Deﬁnition 4. A black point is deleted by the parallel

reduction AS if it satisﬁes all conditions of Theorem 7

(i.e., asymmetric point-based condition for topology-

preserving parallel reductions).

The support of an image operator O is the minimal

set of points whose values determine whether a point

is changed by O (Hall, 1996). The support of the par-

allel reduction S contains 84 points (see Fig. 7), and

the count of points is 64 in the support of the parallel

reduction AS (see Fig. 8).

Figure 7: The 84 points marked ‘⋆’ are in the symmetric

support of reduction S . Note that elements of the N

(p)

are colored cyan, and points denoted ‘’ are not elements

of the FCC grid.

By Theorems 6 and 7, it is obvious that both

derived parallel reductions S and AS are topology-

preserving. It can be readily seen that reduction AS

can delete more points from a picture than reduction

S does. Figure 9 illustrates the difference between the

two derived reductions.

Figures 10–12 give three illustrative examples of

the otherness of S and A S, in which these reductions

are repeated until no voxels are deleted. Numbers in

parentheses are the counts of voxels in the original

objects and the produced residues.

We can state that the iterated asymmetric reduc-

tion A S could extract the topological kernel from all

Topology-Preserving Reductions on (18,12) Pictures of the Face-Centered Cubic Grid

259

Figure 8: The 64 points in the asymmetric support of re-

duction AS marked ‘⋆’. The elements of the sets { q | q ∈

(p), p ≺ q} and { r | r ∈ N

(p), r ≺ p} are colored

cyan and red, respectively. Note that grid points marked ‘

’

are just taken into consideration by the symmetric reduction

S (see Fig. 7), and the points denoted ‘’ are not in the FCC

grid.

original object

reduced by S reduced by AS

Figure 9: A 9 ×9 ×9 cube containing 365 voxels and its

reduced versions produced by the two derived reductions.

The symmetric reduction S and the asymmetric reduction

AS deleted 122 and 154 voxels, respectively. Note that red

voxels are not deleted by S , while AS managed to remove

them.

the three test objects (see Figs. 10–12). A topological

kernel of an object is a minimal set of points that is

topologically equivalent (Kong and Rosenfeld, 1989)

to the original object. It can be readily seen that there

is no simple point in a topological kernel.

Due to the conditions of Theorem 6 the iterated

symmetric reduction A S may produce 2-voxel wide

line segments with a number or simple points in them.

Thus the iterated symmetric reduction AS cannot ex-

tract topological kernels.

It is worthy of note that since the 9 ×9 ×9 cube

(see Fig. 9) is symmetric, and it is free from holes and

cavities (i.e., internal bubbles), both iterated reduc-

original object (996)

shrunk by S (52) shrunk by AS (36)

Figure 10: A 26 ×25 ×7 image of a torus and its shrunk

versions produced by the two iterated reductions.

original object (1505)

shrunk by S (48) shrunk by AS (33)

Figure 11: A 23×27×9 image of a letter “A” and its shrunk

versions produced by the two iterated reductions.

tions can shrink that object to an isolated black point

(i.e., a singleton object).

ICPRAM 2023 - 12th International Conference on Pattern Recognition Applications and Methods

260

original object (37124)

shrunk by S (316) shrunk by AS (188)

Figure 12: A 45×45×45 image of a cube with two holes

and its shrunk versions produced by the two iterated reduc-

tions.

6 CONCLUSIONS

In this paper, we gave two conﬁguration-based and

two point-based sufﬁcient conditions for topology-

preserving parallel reductions acting on (18, 12) pic-

tures of the FCC grid. The conﬁguration-based con-

ditions provide methods of verifying that formerly

constructed parallel reductions preserve the topology.

Our point-based conditions directly provide deletion

rules of topology-preserving parallel reductions, and

allow us to construct topologically correct parallel

thinning algorithms.

As a future work we intend to combine our

point-based conditions with parallel thinning strate-

gies and geometric constraints to generate a family of

topology-preserving parallel thinning algorithms on

the FCC grid.

ACKNOWLEDGEMENTS

The research leading to these results has received

funding from the national project TKP2021-NVA-

09. Project no. TKP2021-NVA-09 has been imple-

mented with the support provided by the Ministry of

Innovation and Technology of Hungary from the Na-

tional Research, Development and Innovation Fund,

ﬁnanced under the TKP2021-NVA funding scheme.

REFERENCES

Comi´c, L. and Magillo, P. (2020). On Hamiltonian cycles

in the FCC grid. Computers & Graphics, 89:88–93.

Comi´c, L. and Nagy, B. (2016). A topological 4-coordinate

system for the face centered cubic grid. Pattern

Recognition Letters, 83:67–74.

Edelsbrunner, H., Iglesias-Ham, M., and Kurlin, V. (2015).

Relaxed disk packing. In Proc. 27th Canadian Con-

ference on Computational Geometry, CCCG 2015.

Queen’s University, Ontario, Canada.

Gastineau, N. and Togni, O. (2021). Coloring of the d

power of the face-centered cubic grid. Discussiones

Mathematicae - Graph Theory, 41:1001–1020.

Gau, C. J. and Kong, T. Y. (1999). Minimal nonsimple sets

of voxels in binary images on a face-centered cubic

grid. Int. J. Pattern Recognition and Artiﬁcial Intelli-

gence, 13:485–502.

Hall, R. W. (1996). Parallel connectivity-preserving thin-

ning algorithms. In Kong, T. Y. and Rosenfeld, A.,

editors, Topological algorithms for digital image pro-

cessing, pages 145–179. Elsevier Science.

Herman, G. T. (1998). Geometry of digital spaces.

Birkh¨auser.

Kardos, P. and Pal´agyi, K. (2015). Topology preservation

on the triangular grid. Annals of Mathematics and Ar-

tiﬁcial Intelligence, 75:53–68.

Kong, T. Y. (1995). On topology preservation in 2-d and

3-d thinning. Int. J. Pattern Recognition and Artiﬁcial

Intelligence, 9:813–844.

Kong, T. Y. and Rosenfeld, A. (1989). Digital topology:

Introduction and survey. Computer Vision, Graphics,

and Image Processing, 48:357–393.

Koshti, G., Biswas, R., Largeteau-Skapin, G., Zrour, R.,

Andres, E., and Bhowmick, P. (2018). Sphere con-

struction on the FCC grid interpreted as layered

hexagonal grids in 3D. In Barneva, R., Brimkov, V.,

and Tavares, J., editors, Combinatorial Image Anal-

ysis, pages 82–96. Springer International Publishing,

LNCS, vol 11255.

Pal´agyi, K., N´emeth, G., and Kardos, P. (2012). Topol-

ogy preserving parallel 3d thinning algorithms. In

Brimkov, V. E. and Barneva, R. P., editors, Digital Ge-

ometry Algorithms. Theoretical Foundations and Ap-

plications to Computational Imaging, pages 165–188.

Springer.

R´acz, G. F. and Cs´ebfalvi, B. (2018). Cosine-weighted B-

spline interpolation on the face-centered cubic lattice.

Computer Graphics Forum, 37:503–511.

Ronse, C. (1988). Minimal test patterns for connectivity

preservation in parallel thinning algorithms for binary

digital images. Discrete Applied Mathematics, 21:67–

79.

Saha, P. K., Borgefors, G., and Sanniti di Baja, G. (2016).

A survey on skeletonization algorithms and their ap-

plications. Pattern Recognition Letters, 76:3–12.

Strand, R. and Stelldinger, P. (2007). Topology preserving

marching cubes-like algorithms on the face-centered

cubic grid. In 14th Int. Conference on Image Analysis

& Processing, pages 781–786.

Topology-Preserving Reductions on (18,12) Pictures of the Face-Centered Cubic Grid

261