Subﬁeld-based Parallel Kernel-thinning Algorithms on the BCC Grid

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, Kernel-thinning, BCC Grid, Topology-preservation.

Abstract:

Kernel-thinning is a widely used technique for extracting the topological kernel from a digital object (i.e.,

producing a minimal structure that is topologically equivalent to the original elongated object). In this paper,

two subﬁeld-based parallel kernel-thinning algorithms acting on the non-standard body centered cubic (BCC)

grid are presented. Our algorithms combine a sufﬁcient condition for topology preservation with two types of

partitionings of the BCC grid, thus both algorithms are topology-preserving. According to our best knowledge,

the reported algorithms are the very ﬁrst parallel thinning algorithms on the BCC grid.

1 INTRODUCTION

A binary digital picture (picture in short) on a discrete

space is composed of black or white points (Kong

and Rosenfeld, 1989). Skeletonization provides fre-

quently applied region-based shape descriptors which

represent the general shape of binary objects that are

present in images (Saha et al., 2016; Saha et al., 2017;

Siddiqi and Pizer, 2008). In 3D, there are three types

of skeleton-like features: the centerline, the medial

surface, and the topological kernel. The centerline

is a line-like 1D representation of objects that cap-

tures the part-whole structure of the object to be de-

scribed (Cornea et al., 2007; Sobiecki et al., 2014).

In many applications (Saha et al., 2016), it is a con-

cise representation of tubular and tree-like 3D objects.

The medial surface provides an approximation to the

continuous 3D skeleton, since it can contain 2D sur-

face patches. A topological kernel of a 3D object is a

minimal set of points that is topologically equivalent

(Kong and Rosenfeld, 1989) to the original object. It

is fairly useful in representing or checking the topo-

logical structure of the object to be processed. Note

that a topological kernel of a 3D object is an isolated

point if and only if it does not contain any holes (that

donuts have) nor cavities (i.e., bubbles). Otherwise,

topological kernels are formed by 1-point thick closed

curves and/or 1-point thick closed surfaces.

Several approaches have been proposed for pro-

ducing skeleton-like features from (segmented) bi-

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

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

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

nary objects. Some authors presented comprehen-

sive and concise surveys (Saha et al., 2016; Saha

et al., 2017; Siddiqi and Pizer, 2008; Sobiecki et al.,

2014; Tagliasacchi et al., 2016). Thinning is one of

the major skeletonization techniques (Hall, 1996). It

is an iterative object-reduction process for producing

skeleton-like features in a topology-preserving way:

the outmost layer of an object is deleted, and the

entire process is repeated until stability is reached.

The topology-oriented thinning pays less attention

to the metric properties of the object to be repre-

sented, since the invariance under arbitrary rotation

angles or scaling factors is not fulﬁlled. In spite of

these drawbacks, our attention has been focused on

thinning, since thinning is the fastest skeletonization

method, it can be implemented easily, it can pro-

duce all types of skeleton-like features, the topology-

preservation can be guaranteed, and thinning provides

practically exquisite descriptors for a number of ap-

plications. There are three kinds of 3D thinning al-

gorithms according to the three types of skeleton-like

shape features: curve-thinning algorithms are used to

extract centerlines, surface-thinning algorithms pro-

duce medial surfaces, while kernel-thinning or reduc-

tive shrinking ones are capable of extracting topolog-

ical kernels (Hall et al., 1996).

A parallel reduction transforms a picture only by

changing some set of black points to white ones si-

multaneously, which is referred to as deletion (Hall,

1996). Parallel thinning algorithms are composed of

parallel reductions, and they fall into three major cat-

egories: fully parallel, subiteration-based (or direc-

tional), and subﬁeld-based (Hall, 1996). Fully par-

288

Karai, G., Kardos, P. and Palágyi, K.

Subﬁeld-based Parallel Kernel-thinning Algorithms on the BCC Grid.

DOI: 10.5220/0010831000003122

In Proceedings of the 11th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2022), pages 288-295

ISBN: 978-989-758-549-4; ISSN: 2184-4313

allel algorithms apply the same parallel reduction in

each thinning phase; in subiteration-based algorithms

a cycle of a small set of parallel reductions are as-

signed to the selected kinds of deletion directions, and

only border points of a certain kind can be deleted at

a subiteration; subﬁeld-based algorithms partition the

given digital space into k ≥ 2 subsets which are al-

ternatively activated, and only some points in the ac-

tive subﬁeld can be deleted simultaneously. Similarly

to the directional approach, an iteration step of a k-

subﬁeld algorithm is composed of k subcycles (i.e.,

parallel reductions).

Existing 3D thinning algorithms act on the con-

ventional 3D cubic grid, in which each point is associ-

ated with an element of Z

(i.e., a point in the 3D Eu-

clidean space with integer coordinates), and its voxel-

representation contains unit cubes. According to our

best knowledge, no one proposed kernel-thinning on

the body-centered cubic (BCC) grid. The voxel-

representations associated with this non-standard grid

contain truncated octahedra. The importance of the

BCC grid shows an upward tendency due to its advan-

tages of geometric and topologic properties (

Comi

and Nagy, 2016; Cs

ebfalvi, 2013; Matej and Lewitt,

1995; Strand, 2004; Strand and Nagy, 2008; Theussl

et al., 2001).

In this paper, the very ﬁrst topology-preserving

parallel kernel-thinning algorithms acting on the BCC

grid are presented. Both of the proposed algorithms

fall into the category of subﬁeld-based.

The rest of this paper is organized as follows: Sec-

tion 2 gives an outline of the key concepts of digital

topology and the relevant results are described. Then

in Section 3, two novel kernel-thinning algorithms are

proposed. In Section 4 results on some test images

produced by our algorithms are given. Finally, we

round off this paper with some concluding remarks.

2 BASIC NOTIONS AND RESULTS

Next, we deﬁne the key concepts of digital topology

as reviewed in (Kong and Rosenfeld, 1989).

A (14, 14) picture on the BCC grid is a quadruple

(B, 14, 14, B), where an element of B is assigned to

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

point in B \B is said to be a white point; The same ad-

jacency relation called 14-neighborhood is assigned

to the sets of black and white points. Let N

(p) de-

note the set of points that are 14-adjacent to p, see

Fig. 1.

Since the studied adjacency relation is symmetric,

its reﬂexive-transitive closure forms an equivalence

relation, and the generated equivalence classes of a

Figure 1: The studied adjacency relation on B (left). The

14 points marked ‘•’ form the set N

(p). (Note that un-

marked elements in Z

are not points in B.) The voxel-

representation of N

(p), where each voxel is a truncated

octahedron (right).

set of points are called components. A black com-

ponent or an object is a 14-component of B, while a

white component is a 14-component of B \ B.

A point p ∈ B is an interior point for B, if all points

being 14-adjacent to p are in B (i.e., N

(p) ⊂ B), p is

called a border point if it is not an interior point, and

p is said to be an isolated point if it forms a singleton

object (i.e., N

(p) ∩ B =

0).

Thinning algorithms, composed of reductions, are

required to preserve topology (Kong, 1995). A reduc-

tion in 2D does not preserve topology if any object

in the input picture is split (into several objects) or is

completely deleted, any white component in the input

picture is merged with another white component, or

a white component is created where there was none

in the input picture. There is an additional concept

called hole in 3D pictures. Holes (which donuts have)

are formed from white points, but they are not white

components (Kong and Rosenfeld, 1989). Topology

preservation in 3D implies that eliminating or creating

any hole is not allowed.

A black point is said to be simple if its deletion is

a topology-preserving reduction. Now we will make

use of the following characterization of simple points:

Theorem 1. (Strand and Brunner, 2006) A point p ∈

B in picture (B, 14, 14, B) is simple if and only if the

following conditions hold:

1. N

(p) ∩ B contains exactly one component.

2. N

(p) \ B contains exactly one component.

It is an easy consequence of Theorem 1 that only non-

isolated border points may be simple, and the simple-

ness is a local property (i.e., it can be decided by ex-

amining the points that are 14-adjacent to the given

black point). Figure 2 gives four illustrative examples

of simple and non-simple points.

Parallel reductions delete a set of points and not

just a single black point. Thus we need to consider

what is meant by topology preservation when a num-

Subﬁeld-based Parallel Kernel-thinning Algorithms on the BCC Grid

289

ber of points are deleted at a time. One of the au-

thors established the following sufﬁcient conditions

for topology-preserving reductions:

Theorem 2. (Kardos, 2021) A reduction is topology-

preserving in picture (B, 14, 14, B) if the following

conditions hold:

1. Only simple points are deleted.

2. If two 14-adjacent points p and q are deleted, q is

simple in picture (B, 14, 14, B \ {p}).

3. If three mutually 14-adjacent points are deleted,

they can be arranged in a sequence hp, q, ri such

that p is simple in picture (B, 14, 14, B), q is sim-

ple in picture (B, 14, 14, B \ {p}), and r is simple

in picture (B, 14, 14, B \ {p, q}).

4. No object formed by four mutually 14-adjacent

black points is deleted completely.

Figure 2: Examples of simple and non-simple points. The

positions marked ‘•’ and ‘◦’ represent black and white

points, respectively. Black point p is simple only in the top

left conﬁguration. In the top right example, p is an isolated

black point, thus Condition 2 of Theorem 1 is not hold. In

the bottom left conﬁguration, we can ﬁnd two components

in N

(p) ∩ B, hence Condition 1 of Theorem 1 is violated.

The bottom right conﬁguration depicts a case where there

exist two 14-components in N

(p)\ B, thus Condition 2 of

Theorem 1 does not hold.

In Section 3, two subﬁeld-based parallel kernel-

thinning algorithms are reported. We propose the par-

titions of B into four and eight subﬁelds as shown in

Fig. 3 and Fig. 4, respectively.

Let us state now an important property of these

partitionings:

Proposition 1. If p ∈ S

(i) (k = 4, 8; i = 0, 1, . . . , k −

1) and q ∈ N

(p), q 6∈ S

(i).

It is obvious by careful examination of Fig. 3 and

Fig. 4.

By Proposition 1, Theorem 2 can by simpliﬁed for

reductions of subﬁeld-based thinning algorithms:

Theorem 3. A reduction that deletes a subset of

(i) ∩ B (k = 4, 8; i = 0, 1, . . . , k − 1) from picture

(B, 14, 14, B) is topology-preserving if only simple

points are deleted.

Proof. It is obvious that Condition 1 of Theorem 2 is

satisﬁed.

With the help of Proposition 1, it can be readily

seen that there is no mutually 14-adjacent pair (triplet

and quadruple) of points in the same subﬁeld accord-

ing to the proposed partitionings. Thus the last three

conditions of Theorem 2 are irrelevant here.

Figure 3: Partition of B into four subﬁelds. All points

marked ‘i’ are in subﬁeld S

(i) (i = 0, 1, 2, 3). (Note that

unmarked elements in Z

are not points in B).

Figure 4: Partition of B into eight subﬁelds. All points

marked ‘i’ are in subﬁeld S

(i) (i = 0, 1, . . . , 7). (Note that

unmarked elements in Z

are not points in B).

ICPRAM 2022 - 11th International Conference on Pattern Recognition Applications and Methods

290

3 NEW SUBFIELD-BASED

PARALLEL THINNING

ALGORITHMS

In this section, two parallel algorithms are reported:

• SF-4-TK: 4-subﬁeld kernel-thinning algorithm,

• SF-8-TK: 8-subﬁeld kernel-thinning algorithm.

Algorithm 1 gives the two kernel-thinning algorithms.

Algorithm 1: Algorithm SF-k-TK (k = 4, 8).

Input: picture (B, 14, 14, X)

Output: picture (B, 14, 14, Y )

// initialize resulted black points

Y ← X

repeat

// initialize deleted points

D ←

// k subiterations

for i ← 0 to k − 1 do

// collect deletable points

D(i) ← { p | p ∈ S

(i) ∩Y and simple }

// parallel deletion

Y ← Y \ D(i)

// collect deleted points

D ← D ∪ D(i)

until D =

In Algorithm 1 the kernel of the repeat cycle cor-

responds to one iteration step that comprises k (k =

4, 8) subiterations (i.e., parallel reductions) in which

the k subﬁelds (see Fig. 3 and Fig. 4) are alternatively

activated. The thinning process is terminated if no

points are deleted within an iteration step (i.e., D =

0).

Note that a general and computationally efﬁcient

implementation scheme for parallel thinning algo-

rithms was proposed in (Pal

agyi, 2008). This scheme

takes advantage of the fact that all thinning algorithms

may delete only border points. Thus we do not have

to examine the deletability of interior points, and the

repeated scans of the entire array (that stores the ac-

tual picture) can be avoided by using a linked list that

stores the set of border points to be evaluated for pos-

sible deletion in the actual thinning phase.

Notice that only simple points can be deleted in

each subiteration of all the proposed subﬁeld-based

parallel thinning algorithms (see Algorithm 1). Thus

we can state the following:

Theorem 4. Algorithms SF-4-TK and SF-8-TK are

both topology-preserving.

Proof. By Theorem 3, it is obvious that each subit-

eration (i.e., parallel reduction) of our algorithms is

topology-preserving.

Since the proposed subﬁeld-based parallel kernel-

thinning algorithms are composed of topology-

preserving reductions, the entire algorithms are

topology-preserving.

We remark that in our kernel-thinning algorithms

(see Algorithm 1) we do not need to use any geomet-

ric constraint.

Note that we follow a safe technique for design-

ing topology-preserving parallel thinning algorithms:

some sufﬁcient conditions for topology-preserving re-

ductions are combined with parallel thinning strate-

gies and geometric constraints (Pal

agyi et al., 2012).

Lastly, since the output of Algorithm 1 does not

contain any simple point, the presented algorithms are

indeed kernel-thinning ones:

Proposition 2. Algorithms SF-4-TK and SF-8-TK

extract the topological kernel of the input object.

4 RESULTS

The proposed two algorithms were tested on objects

of different shapes. We present ten illustrative exam-

ples below (see Figs. 5-14). The numbers in parenthe-

ses under the original images are the counts of object

points. The pairs of numbers in parentheses under

the produced topological kernels indicate the counts

of object points and the numbers of the required iter-

ations, respectively. Notice that the algorithms trans-

form the synthetic object with no hole in Fig. 5 to

a single voxel, while the topological kernels of ho-

ley objects in Figs. 6-14 are one-voxel thin connected

closed curve segments. Notice that in Fig. 14, our al-

gorithms produced exactly the same skeleton-like fea-

tures, and even the sufﬁcient number of iterations are

equal.

Our implementations were run on a usual desk-

top (HP ProDesk 400 G4; 3.20 GHz Intel Core i5-

6500; Windows 10 x64) and written in C++. The

std::chrono library was used for runtime measure-

ment. Table 1 contains the computation times of our

algorithms for each object shown in Figs. 5-14. Note

that reading the input image and writing the output

image were not considered here. We can observe that

the computational cost does not depend on the num-

ber of subﬁelds, due to the fact that any border point

belongs to exactly one subﬁeld. Thus, any of those

points is examined exactly once in each iteration.

Since the authors do not know any existing algo-

rithm acting on the BCC grid for producing topolog-

ical kernels, we can not compare the results of SF-4-

TK and SF-8-TK with competing methods. Note that

topological kernels could be produced by sequential

Subﬁeld-based Parallel Kernel-thinning Algorithms on the BCC Grid

291

deletion of simple points and by various skeletoniza-

tion approaches, but these strategies are out of scope

of this work.

Table 1: Computation times (in millisec.) of Algorithm 1.

Test object SF-4-TK SF-8-TK

Syntetic object 0.993 0.992

Torus 1.994 1.995

Gear 36.869 36.889

Hand 31.947 32.897

Letter A 9.942 9.954

‘Fertility’ 29.908 29.920

Helicopter 38.890 38.895

Cube with 1 hole 37.866 37.890

Cube with 2 holes 33.914 33.909

Cube with 3 holes 31.891 31.902

(2 806)

(1, 8) (1, 10)

Figure 5: A 32 × 32 × 32 image of a synthetic object

(top) and its topological kernels produced by algorithms

SF-4-TK (bottom left) and SF-8-TK (bottom right).

(10 028)

(112, 7) (84, 8)

Figure 6: A 64 × 64 × 19 image of a synthetic object

(top) and its topological kernels produced by algorithms

SF-4-TK (bottom left) and SF-8-TK (bottom right).

(149 142)

(218, 25) (234, 27)

Figure 7: A 45 × 191 × 191 image of a gear (top) and

its topological kernels produced by algorithms SF-4-TK

(bottom left) and SF-8-TK (bottom right).

ICPRAM 2022 - 11th International Conference on Pattern Recognition Applications and Methods

292

(113 751)

(151, 51) (165, 43)

Figure 8: A 191 × 96 × 114 image of a hand (top) and

its topological kernels produced by algorithms SF-4-TK

(bottom left) and SF-8-TK (bottom right).

(41 964)

(109, 17) (109, 20)

Figure 9: A 100 × 100 × 40 image of a letter (top) and

its topological kernels produced by algorithms SF-4-TK

(bottom left) and SF-8-TK (bottom right).

(100 188)

(471, 18) (459, 17)

Figure 10: A 138 × 70 × 189 image of ‘fertility’ (top) and

its topological kernels produced by algorithms SF-4-TK

(bottom left) and SF-8-TK (bottom right).

(68 432)

(304, 85) (272, 72)

Figure 11: A 381 × 103 × 255 image of a helicopter

(top) and its topological kernels produced by algorithms

SF-4-TK (bottom left) and SF-8-TK (bottom right).

Subﬁeld-based Parallel Kernel-thinning Algorithms on the BCC Grid

293

(178 808)

(184, 20) (156, 21)

Figure 12: A 93 × 93 × 93 image of a holey cube

(top) and its topological kernels produced by algorithms

SF-4-TK (bottom left) and SF-8-TK (bottom right).

(163 928)

(330, 17) (320, 18)

Figure 13: A 93× 93 × 93 image of a holey cube with more

holes (top) and its topological kernels produced by algo-

rithms SF-4-TK (bottom left) and SF-8-TK (bottom right).

(149 048)

(368, 11) (368, 11)

Figure 14: A 93 × 93 × 93 image of a cube with even more

holes (top) and its topological kernels produced by algo-

rithms SF-4-TK (bottom left) and SF-8-TK (bottom right).

5 CONCLUSIONS

In this paper, the very ﬁrst 3D kernel-thinning algo-

rithms on the BCC grid are proposed. These par-

allel algorithms fall into the subﬁeld-based category,

and they can produce topological kernels. The topo-

logical correctness of our algorithm is guaranteed by

the safe designing technique: a sufﬁcient condition

for topology-preserving reductions is combined with

a parallel thinning strategy.

In a future work, we plan to construct new sym-

metric and asymmetric point-based sufﬁcient condi-

tions for topology preservation. Combining these

conditions with all types of parallel thinning methods

and various geometric constraints yields a variety of

parallel 3D curve-, surface-, and kernel-thinning al-

gorithms acting on the BCC grid.

REFERENCES

Comi

c, L. and Nagy, B. (2016). A combinatorial coordinate

system for the body-centered cubic grid. Graphical

Models, 87:11–22.

Cornea, N., Silver, D., and Min, P. (2007). Curve-skeleton

properties, applications, and algorithms. IEEE Trans-

actions on Visualization and Computer Graphics,

13:530–548.

ebfalvi, B. (2013). Cosine-weighted b-spline interpola-

tion: A fast and high-quality reconstruction scheme

for the body-centered cubic lattice. IEEE Transactions

ICPRAM 2022 - 11th International Conference on Pattern Recognition Applications and Methods

294

on Visualization and Computer Graphics, 19:1455–

1466.

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.

Hall, R. W., Kong, T. Y., and Rosenfeld, A. (1996). Shrink-

ing binary images. In Kong, T. Y. and Rosenfeld, A.,

editors, Topological algorithms for digital image pro-

cessing, pages 31–98. Elsevier Science.

Kardos, P. (2021). Topology preservation on the bcc grid.

Journal of Combinatorial Optimization.

Kong, T. Y. (1995). On topology preservation in 2-d and 3-d

thinning. International Journal of 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.

Matej, S. and Lewitt, R. M. (1995). Efﬁcient 3d grids

for image reconstruction using spherically-symmetric

volume elements. IEEE Transactions on Nuclear Sci-

ence, 42:1361–1370.

Pal

agyi, K. (2008). A 3d fully parallel surface-thinning

algorithm. Theoretical Computer Science, 406:119–

135.

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.

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.

Saha, P. K., Borgefors, G., and Sanniti di Baja, G., editors

(2017). Skeletonization: Theory, methods and appli-

cations. Academic Press.

Siddiqi, K. and Pizer, S., editors (2008). Medial Represen-

tations – Mathematics, Algorithms and Applications,

Computational Imaging and Vision, vol. 37. Springer,

New York.

Sobiecki, A., Jalba, A., and Telea, A. (2014). Comparison

of curve and surface skeletonization methods for voxel

shapes. Pattern Recognition Letters, 47:147–156.

Strand, R. (2004). Surface skeletons in grids with non-cubic

voxels. In Proc. of the 17th International Conference

on Pattern Recognition, ICPR 2004, pages 548–551.

Cambridge.

Strand, R. and Brunner, D. (2006). Simple points on the

body-centered cubic grid. Technical Report 42, Cen-

tre for Image Analysis, Uppsala University, Uppsala,

Sweden.

Strand, R. and Nagy, B. (2008). Weighted neighbour-

hood sequences in non-standard three-dimensional

grids - metricity and algorithms. In Proc. of the 14th

IAPR International Conference on Discrete Geometry

for Computer Imagery, DGCI 2008, pages 201–212.

Springer, LNCS, vol 4992.

Tagliasacchi, A., Delame, T., Spagnuolo, M., Amenta, N.,

and Telea, A. (2016). 3d skeletons: A state-of-the-

art report. In Proc. of the 37th Annual Conference

of the European Association for Computer Graphics,

EG’16, pages 573–597.

Theussl, T., M

oller, T., and Gr

olle, M. E. (2001). Opti-

mal regular volume sampling. In Proc. Visualization,

VIS’01, pages 91–98.

Subﬁeld-based Parallel Kernel-thinning Algorithms on the BCC Grid

295