1-Attempt 4-Cycle Parallel Thinning Algorithms

K´alm´an Pal´agyi

and G´abor N´emeth

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

Keywords:

Shape Representation, Skeletonization, Thinning, Digital Topology.

Abstract:

Thinning is a frequently applied skeletonization technique. It is an iterative object-reduction in a topology-

preserving way: the outmost layer of an object is deleted, and the entire process is repeated until stability is

reached. In the case of an 1-attempt thinning algorithm, if a border pixel is not deleted in the very ﬁrst time, it

cannot be deleted in the remaining phases of the thinning process. This paper shows that two 4-cycle parallel

2D thinning algorithms (i.e., one subiteration-based and one subﬁeld-based) are 1-attempt. In addition, we

illustrate that both algorithms are considerably faster if we know that they fulﬁll the 1-attempt property.

1 INTRODUCTION

A binary digital picture (picture in short) on the 2D

square grid is composed of black or white unit squares

that are called as pixels. A parallel reduction trans-

forms a picture only by changing some set of black

pixels to white ones at a time, which is referred to as

deletion (Hall, 1996).

Skeletonization in 2D provides centerline that is

a compact representation of a binary object, and it is

used in various image processing and computer vi-

sion applications (Saha et al., 2017; Siddiqi and Pizer,

2008).

Thinning is an iterative skeletonization technique:

border pixels of a binary object that satisfy certain

topological and geometric constraints are deleted, and

this process is repeated until no pixels are deleted

(i.e., the centerline is reached) (Hall, 1996; Kong and

Rosenfeld, 1989). Parallel thinning algorithms are

composed of parallel reductions, and they fall into

three major categories: fully parallel, subiteration-

based (or directional), and subﬁeld-based (Hall, 1996;

N´emeth and Pal´agyi, 2011). Fully parallel algorithms

apply the same parallel reduction in each iteration; in

subiteration-based algorithms a cycle of d ≥ 2 paral-

lel reductions are assigned to the selected d kinds of

deletion directions, and only border pixels of a certain

kind can be deleted at a subiteration; subﬁeld-based

algorithms partition Z

into s ≥ 2 subsets which are

alternatively activated, and only some pixels in the

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

https://orcid.org/0000-0002-4118-5019

active subﬁeld can be deleted simultaneously. Sim-

ilarly to the directional approach, an iteration step of

an s-subﬁeld algorithm is composed of s parallel re-

ductions. That is why n-subiteration and n-subﬁeld

algorithms are referred to as n-cycle algorithms.

In the conventional implementation of thinning

algorithm, all border pixels are examined for possi-

ble deletion in each thinning phase. Thinning is the

fastest skeletonization approach, however, the con-

ventional scheme is a rather time-consuming for ob-

jects that also contain ‘thin’ and ‘thick’ parts, since

some pixels of the produced centerline ‘get free’

within ‘a few’ (sub)iterations, but the iterative object

reduction process is to be continued until no pixels

are deleted. That is why the authors of this paper in-

troduced the concept of 1-attempt thinning (Pal´agyi

and N´emeth, 2021a). In a 1-attempt thinning algo-

rithm, if a border pixel cannot be deleted in the very

ﬁrst examination for possible deletion, this pixel re-

mains unchanged in the remaining thinning phases

(i.e., it is an element of the produced centerline). In

our previous papers we constructed a new fully paral-

lel thinning algorithm, and proved that it is 1-attempt

(Pal´agyi and N´emeth, 2021a); we managed to prove

that an existing 2-subﬁeld parallel thinning algorithm

(N´emeth and Pal´agyi, 2011) is also 1-attempt (Pal´agyi

and N´emeth, 2021b); we showed that the thinning al-

gorithms fulﬁlling the 1-attempt property can be im-

plemented with a remarkable speed up (Pal´agyi and

N´emeth, 2021a; Pal´agyi and N´emeth, 2021b).

In this work, our attention is focussed on 4-cycle

parallel thinning: we prove that a 4-subiteration (di-

rectional) and a 4-subﬁeld thinning algorithm are 1-

Palágyi, K. and Németh, G.

1-Attempt 4-Cycle Parallel Thinning Algorithms.

DOI: 10.5220/0010819700003122

In Proceedings of the 11th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2022), pages 229-236

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

229

Figure 1: The two kinds of adjacency relations studied on

the square grid (a). The four pixels p

, p

, and p

are

4-adjacent to the central pixel p. These four pixels and the

four ones marked ‘•’ are 8-adjacent to p.

Partition of Z

into four subﬁelds (b). All pixels marked ‘i’

are in subﬁeld S

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

attempt, and show that their 1-attempt property is use-

ful (i.e., it makes computationally efﬁcient implemen-

tation schemes possible).

2 BASIC NOTIONS AND RESULTS

Next, we apply the fundamental concepts of digital

topology as reviewed in (Kong and Rosenfeld, 1989).

An (8,4) picture on the 2D square grid is a

quadruple (Z

,8, 4,B), where an element of Z

is as-

signed to each pixel (i.e., unit square); B ⊆ Z

denotes

the set of black pixels; each pixel in Z

\ B is said to

be a white pixel; adjacency relations 8 and 4 are as-

signed to B and Z

\ B, respectively. Two pixels (i.e.,

unit squares) are 4-adjacent if they share an edge, and

they are 8-adjacent if those pixels share an edge or a

vertex, see Fig. 1a. Let N(p) denote the set of (eight)

pixels that are 8-adjacent to p.

Since both studied relations are symmetric, their

reﬂexive-transitive closure form equivalence rela-

tions, and the generated equivalence classes are called

components. A black component or an object is an

8-component of B, while a white component is a 4-

component of Z

\ B.

A pixel p ∈ B is an interior pixel for B, if all pixels

being 4-adjacent to p are in B, p is called a border

pixel if it is not an interior pixel, and p is said to be an

isolated pixel if it forms a singleton object (i.e., it is

not 8-adjacent to a black pixel). Here we distinguish

four types of border pixels: a black pixel p is called

an i-border pixel (i = 0, 1,2, 3) in an (8, 4) picture, if

the pixel marked p

in Fig. 1a is white.

Thinning algorithms preserve some (border) pix-

els that provide relevant geometric information with

respect to the shape of the object. Most of existing

2D thinning algorithms preserve endpixels (i.e., ter-

minating pixels of curves). The two 4-cycle parallel

Figure 2: Examples of the studied two types of endpixels.

In the ﬁrst two conﬁgurations, p is an endpixel of type 1 and

an endpixel of type 2. The third example gives an example

of an endpixel of type 2 (that is not an endpixel of type 1).

thinning algorithms described in Section 3 use the fol-

lowing two types of endpixels (Hall, 1996; N´emeth

and Pal´agyi, 2011):

Deﬁnition 1. A black pixel (in an (8,4) picture) is

an endpixel of type 1 if it is 8-adjacent to exactly one

black pixel.

Deﬁnition 2. A black pixel p (in an (8,4) picture) is

an endpixel of type 2 if it is endpixel of type 1 or it is

8-adjacent to exactly two black pixels q and r, where

q is 4-adjacent to r.

Figure 2 shows some examples of the two types of

endpixels taken into consideration.

A crucial issue in thinning algorithms is to en-

sure topology preservation (Kong, 1995; Kong and

Rosenfeld, 1989). A reduction in a 2D picture is

topology-preservingif each object in the input picture

contains exactly one object in the output picture, and

each white component in the output picture contains

exactly one white component in the input picture. A

single black pixel is simple for a set of black pixels if

its deletion is a topology-preserving reduction. Here

we make use of the following characterization of sim-

ple pixels:

Theorem 1. (Kong and Rosenfeld, 1989) A black

pixel p is simple in an (8,4) picture if all the following

conditions hold:

1. p is a border pixel.

2. There is exactly one black 8-component in N(p).

Note that the simpleness in (8,4) pictures is a lo-

cal property, it can be decided by examining the eight

pixels that are 8-adjacent to the given black pixel,

and only non-isolated border pixels may be simple.

Here we represent non-simple pixels in (8,4) pictures

by the six base matching templates shown in Fig. 3.

All rotated and reﬂected versions of these base tem-

plates match non-simple pixels, too. It can be readily

seen that a pixel is non-simple if at least one template

matches it.

Parallel reductions can delete a set of black pixels.

Hence, we need to consider what is meant by topol-

ogy preservation when a number of black pixels are

deleted simultaneously. Kong gave the following suf-

ﬁcient condition:

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

230

Figure 3: The six base matching templates for character-

izing non-simple pixels in (8, 4) pictures. Notations: each

black template position matches a black pixel; each white

element matches a white pixel; each position depicted in

gray matches either a black or a white pixel.

Theorem 2. (Kong, 1995) A parallel reduction act-

ing on (8,4) pictures is topology preserving, if all the

following conditions hold:

1. Only simple pixels are deleted.

2. For every set {p,q} of two 4-adjacent pixels that

are deleted, p is simple after deletion of q.

3. No object contained in a 2× 2 square is deleted

completely.

Kong’s result provides a method for verifying that

a fully parallel or a subﬁeld-based thinning algorithm

preserves topology. Rosenfeld gave another sufﬁcient

condition that is useful in verifying the topological

correctness of subiteration-based algorithms:

Theorem 3. (Rosenfeld, 1975) A parallel reduction

acting on (8, 4) pictures is topology preserving, if all

the following conditions hold for every pixel p that is

deleted:

1. p is an i-border pixel (i ∈ {0, 1,2, 3}).

2. p is not an endpixel of type 1.

3. There is exactly one black 8-component in N(p).

3 TWO 4-CYCLE PARALLEL

THINNING ALGORITHMS

In this section, we describe two 4-cycle topology-

preserving parallel thinning algorithm acting on (8, 4)

pictures on the square grid. The ﬁrst algorithm SI is

subiteration-based, and the second one SF falls into

the category of subﬁeld-based. Algorithm 1 gives the

studied two algorithms.

In Algorithm 1, the kernel of the repeat cycle

corresponds to one iteration step that comprises four

subiterations (i.e., four parallel reductions). In these

subiterations, algorithm SI considers the four deletion

directions that are speciﬁed by the studied four types

of border pixels, and algorithm SF alternatively acti-

vates the four subﬁelds shown in Fig. 1b.

Algorithm 1: 4-cycle algorithm T (T=SI,SF).

Input: picture (Z

,8,4,X)

Output: picture (Z

,8,4,Y)

Y ← X

repeat

// one iteration step

for i ← 0 to 3 do

-th subiteration

D(i) ← { p | p is T-i-deletable for Y }

Y ← Y \ D(i)

until D(0) ∪ D(1) ∪ D(2) ∪ D(3) =

The description of the two 4-cycle parallel thin-

ning algorithms (i.e., algorithms SI and SF) is com-

pleted by specifying their deletion rules:

Deﬁnition 3. A black pixel p in picture (Z

,8, 4,B) is

SI-i-deletable for B (i = 0,1, 2,3) if all the following

conditions hold:

1. p is an i-border pixel.

2. p is not an endpixel of type 2.

3. p is simple.

A black pixel (in an (8,4) picture) is non-SI-i-

deletable if it is not SI-i-deletable (i = 0, 1,2,3), i.e.,

at least one condition of Deﬁnition 3 is violated.

Deﬁnition 4. A black pixel p in picture (Z

,8, 4,B) is

SF-i-deletable for B (i = 0, 1,2, 3) if all the following

conditions hold:

1. p ∈ S

(i.e., it is in the i-th subﬁeld, see Fig. 1b).

2. p is not an endpixel of type 1.

3. p is simple.

A black pixel (in an (8,4) picture) is said to be

a non-SF-i-deletable pixel if it is not SF-i-deletable

(i = 0, 1,2,3), i.e., at least one condition of Deﬁnition

4 is not satisﬁed.

Since all endpixels of type 1 are also endpixels of

type 2, the following statement holds:

Proposition 1. All endpixels of type 1 are not SI-i-

deletable (i = 0,1, 2,3).

Let us state another obvious proposition:

Proposition 2. If p ∈ S

(i = 0, 1,2, 3) and q ∈ N(p),

q 6∈ S

By Proposition 1 and Proposition 2, it can be read-

ily seen that the parallel reductions that delete SI-

i-deletable or SF-i-deletable pixels satisfy all con-

ditions of Theorem 3. Thus both algorithms are

topology-preserving.

Note that the idea of algorithm SI is originated

from Rosenfeld, but his early algorithm preserves

endpixels of type 1 (Rosenfeld, 1975). Algorithm SF

1-Attempt 4-Cycle Parallel Thinning Algorithms

231

is proposed in (N´emeth and Pal´agyi, 2011), and it is

referred to as SF-4-1.

For reasons of scope, we do not analyze the ge-

ometric properties of the studied two 4-cycle algo-

rithms. Here we present ﬁve illustrative examples of

the centerlines produced by algorithms SI and SF, see

Fig. 4-8. The pairs of numbers in parentheses are the

counts of object pixels in the produced centerlines and

the number of the required iterations.

SI (39685, 107) SF (39824, 107)

Figure 4: Centerlines produced by the studied algorithms

superimposed on a 730× 820 image of a ship. The original

image contains 214310 object pixels.

SI (17128, 123) SF (17366, 123)

Figure 5: Centerlines produced by the studied algorithms

superimposed on a 730 × 730 image of the yin and yang

symbol. The original image contains 190246 object pixels.

By Figs. 4-8, we can state that both algorithms SI

and SF produce one pixel width and well-positioned

centerlines.

4 FULFILLING THE 1-ATTEMPT

PROPERTY

In this section, we show that the studied algorithms SI

and SF fulﬁll the 1-attempt property.

In order to prove that both algorithms are 1-

attempt, we shall state two lemmas, introduce a new

concept, and examine the matching templates shown

in Fig. 9.

Lemma 1. Let p be a black pixel (in an (8,4) pic-

ture) that is matched by the template shown in Fig. 9a

(or its rotated versions). Then p cannot be deleted by

algorithms SI and SF.

Proof. By Deﬁnition 3, Deﬁnition 4, and Proposition

1, only simple pixels that are not endpixels of type 1

SI (10092, 61) SF (10104, 61)

Figure 6: Centerlines produced by the studied algorithms

superimposed on a 730× 795 image of a spider. The origi-

nal image contains 58557 object pixels.

SI (7555, 175) SF (7839, 175)

Figure 7: Centerlines produced by the studied algorithms

superimposed on a 990×804 image of a jellyﬁsh. The orig-

inal image contains 214514 object pixels.

SI (7652, 186) SF (7839, 186)

Figure 8: Centerlines produced by the studied algorithms

superimposed on a 640× 2200 image of a helicopter. The

original image contains 399984 object pixels.

Figure 9: Templates (conﬁgurations) associated with

Lemma 1 (a) and Lemma 2 (b).

can be deleted by algorithms SI and SF.

Since {q} is a black 8-component in N(p), pixel

p is simple (by Theorem 1) iff all the ﬁve pixels a, b,

c, d, and e in N(p) \ {q} are white. In this case p is

an endpixel of type 1. Thus p cannot be deleted by

algorithms SI and SF.

Lemma 2. Let p be a black pixel (in an (8, 4) pic-

ture) that is matched by the template shown in Fig. 9b

(or its rotated versions). Then p cannot be deleted by

algorithms SI and SF.

Proof. By Deﬁnition 3, Deﬁnition 4, and Proposition

1, only simple pixels that are not endpixels of type 1

can be deleted by algorithms SI and SF.

Since {q} is a singleton black 8-component in

N(p), pixel p is simple (by Theorem 1) iff all the three

pixels a, b, and c in N(p) \{q} are white. In this case

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

232

p is an endpixel of type 1. Thus p cannot be deleted

by algorithms SI and SF.

Now let us introduce a new concept.

Deﬁnition 5. Let p ∈ B be a non-simple pixel in pic-

ture (Z

,8, 4,B). The set of pixels S ⊆ N(p) ∩ B is a

simpliﬁer set (associated to p) if pixel p is simple in

picture (Z

,8, 4,B\ S).

For the sake of brevity in the following we will

refer to positions in the matching templates shown in

Fig. 3 as pixels in (8,4) pictures.

Proposition 3. Let p be a non-simple border (black)

pixel (in an (8, 4) picture) that is matched by template

(see Fig. 3). Then p cannot be deleted by algo-

rithms SI and SF.

Proof. Since p is an isolated pixel (i.e., there is no

black pixel in N(p)), there is no simpliﬁer set associ-

ated to p. Thus p remains a non-simple black pixel.

By Deﬁnition 3 and Deﬁnition 4, only simple pix-

els can be deleted by algorithms SI and SF. Thus p

cannot be deleted.

Proposition 4. Let p ∈ B be a non-simple border

(black) pixel in picture (Z

,8, 4,B) that is matched by

template T

(see Fig. 3). Then p cannot be deleted by

algorithms SI and SF.

Proof. We can state that {q} and {s} are the two pos-

sible simpliﬁer sets associated to p. This follows from

a careful examination of T

Since, by Lemma 1, q and s cannot be deleted by

algorithms SI and SF, p remains a non-simple black

pixel.

By Deﬁnition 3 and Deﬁnition 4, only simple pix-

els can be deleted by the studied two algorithms. Thus

p cannot be deleted.

Proposition 5. Let us assume that a non-simple bor-

der (black) pixel p (in an (8,4) picture) is matched by

template T

(see Fig. 3). Then p cannot be deleted by

algorithms SI and SF.

Proof. Since, by Lemma 1, all the two black pixels

(i.e., q and u) and the two potentially black ones (i.e.,

s and w) cannot be deleted by algorithms SI and SF,

p remains a non-simple black pixel.

By Deﬁnition 3 and Deﬁnition 4, only simple pix-

els can be deleted by algorithms SI and SF. Thus p

cannot be deleted.

Proposition 6. Let us assume that a non-simple bor-

der (black) pixel p ∈ B in picture (Z

,8, 4,B) is

matched by template T

(see Fig. 3). Then p cannot

be deleted by algorithms SI and SF.

Proof. It can be readily seen that {s} and S =

{q,u,v, w,x} ∩ B are the two possible simpliﬁer sets

associated to p.

Since, by Lemma 1, pixel s cannot be deleted in

the thinning process of none of the two algorithms SI

and SF, let us examine the deletability of S.

It is obvious that p becomes an endpixel of type 1

after the deletion of S.

By Deﬁnition 3 and Deﬁnition 4, only simple pix-

els that are not endpixels of type 1 can be deleted by

algorithms SI and SF. Thus the simpliﬁed p cannot

be deleted.

Let us summarize the previously stated four

propositions:

Proposition 7. If a non-simple border (black) pixel p

is matched by template T

, T

, or T

in the input or

an interim picture of algorithms SI or SF, p cannot

be deleted in the remaining thinning phases (i.e., it is

an element of the produced centerline).

Let us take the remaining two matching templates

shown in Fig. 3 into consideration.

Proposition 8. Let p ∈ B be a non-simple border

(black) pixel in picture (Z

,8, 4,B) that is matched by

template T

(see Fig. 3). Then exactly one of the two

sets of pixels {q,w, x} ∩ B and {s,t,u} ∩ B is a subset

of all possible simpliﬁer sets associated to p.

This simply follows from Theorem 1 and a careful

examination of template T

Lastly an absolutely obvious statement is made:

Proposition 9. Let us assume that a (non-simple

black) pixel p (in an (8,4) picture) is matched by tem-

plate T

(see Fig. 3). Then p is not a border pixel (i.e.,

it is an interior pixel).

4.1 Examining Algorithm SI

We are now ready to state the following theorem:

Theorem 4. Algorithm SI is 1-attempt.

Proof. Without loss of generality, we can assume that

0-border pixels are examined for possible deletion in

the actual subiteration.

We need to verify that if a 0-border pixel p ∈ B

in picture (Z

,8, 4,B) is non-SI-0-deletable in the ac-

tual subiteration, it remains non-SI-i-deletable in the

remaining thinning phases for any i ∈ {0,1,2, 3}.

Let us assume indirectly that p is SI-i-deletable

for B\ D for some i ∈ {0,1, 2,3} and for some set of

deleted pixels D ⊆ B\ {p}.

Since pixel p is a non-SI-0-deletable (0-border)

pixel for B, at least one of the last two conditions of

Deﬁnition 3 is violated. Consequently,

• p is an endpixel of type 2 for B, or

• p is non-simple for B.

1-Attempt 4-Cycle Parallel Thinning Algorithms

233

Figure 10: Conﬁgurations associated with Theorem 4 and

Theorem 5.

It is obvious that if p is an endpixel of type 2 for

B (i.e., the ﬁrst point holds), p remains an endpixel of

type 2 for B \ D for any D ⊆ B\ {p}. Thus only the

second point is to be examined (i.e., p is non-simple

for B).

According to our assumption, p is deleted in a re-

maining thinning phase, thus it becomes simple for

B \ D. It means that N(p) ∩ D is a simpliﬁer set of

deleted pixels.

Since p is a non-simple pixel for B, at least one

template shown in Fig. 3 matches it. By Proposition 7

and Proposition 9 we can ignore the ﬁve templates T

, T

, and T

. Thus template T

is the only one to

be investigated.

If p is matched by T

, by Proposition 8, ex-

actly one of the two sets of pixels {q, w,x} ∩ B and

{s,t,u} ∩ B is a subset of all possible simpliﬁer sets

associated to p. Consequently,one of {q, w,x}∩B and

{s,t,u} ∩ B is to be completely deleted. Without loss

of generality, we can assume that ({q, w,x} ∩B) ⊆ D.

It is known that black pixel x ∈ D (see Fig. 3), thus

the following four cases (shown in Fig. 10) are to be

checked:

• q 6∈ D,w 6∈ D (see Fig. 10a):

In this case, by Lemma 2, pixel x cannot be

deleted. This is a contradiction as x ∈ D.

• q 6∈ D,w ∈ D (see Fig. 10b):

Then the following points have to be checked:

– If x and w are deleted in different subiterations,

and x is deleted ﬁrst, then, by Lemma 1, pixel

w cannot be deleted. This is a contradiction as

w ∈ D.

– If x and w are deleted in different subiterations,

and w is deleted ﬁrst, then, by Lemma 2, pixel

x cannot be deleted. This is a contradiction as

x ∈ D.

– If x and w are deleted simultaneously (i.e., in

the same subiteration), then

w is not SI-0-deletable, since it is not a 0-

border pixel (i.e., pixel x is black);

x is not SI-1-deletable, since it is not a 1-

border pixel (i.e., pixel p is black);

x is not SI-2-deletable, since it is not a 2-

border pixel (i.e., pixel w is black).

Thus both pixels x and w need to be SI-3-

deletable. Consequently, the two pixels e and

f are white. It can be readily seen by Theorem

1 that pixel x is simple iff pixel d is also white.

In this case, x is an endpixel of type 2. Thus x

is not SI-3-deletable, and we arrive at a contra-

diction.

• q ∈ D,w 6∈ D (see Fig. 10c):

Similarly to the previous case, the following three

points are to be examined:

– If x and q are deleted in different subiterations,

and x is deleted ﬁrst, then, by Lemma 1, pixel

q cannot be deleted. This is a contradiction as

q ∈ D.

– If x and q are deleted in different subiterations,

and q is deleted ﬁrst, then, by Lemma 2, pixel

x cannot be deleted. This is a contradiction as

x ∈ D.

– If x and q are deleted simultaneously (i.e., in the

same subiteration), then

x is not SI-0-deletable, since it is not a 0-

border pixel (i.e., pixel q is black);

x is not SI-1-deletable, since it is not a 1-

border pixel (i.e., pixel p is black);

q is not SI-2-deletable, since it is not a 2-

border pixel (i.e., pixel x is black).

Thus both pixels x and w need to be SI-3-

deletable. Consequently, the two pixels d and

e are white. It can be readily seen by Theorem

1 that pixel x is simple iff pixel f is also white.

In this case, x is an endpixel of type 2. Thus x

is not SI-3-deletable, and we arrive at a contra-

diction.

• q ∈ D,w ∈ D (see Fig. 10d):

In this case, the following two points are to be

checked:

– Let us assume that the three pixels q, x, and w

are deleted in different subiterations.

If just one of the two pixels q and w is deleted

ﬁrst, then we get the previously examined

cases shown in Fig. 10b and Fig. 10c. Thus

we arrive at a contradiction.

If pixel x is deleted ﬁrst, then both pixels q

and w cannot be deleted by Lemma 1. This is

a contradiction as q ∈ D and w ∈ D.

If the two pixels q and w are deleted ﬁrst, then

we get the previously examined case shown in

Fig. 10a. Thus we arrive at a contradiction.

If the two pixels q and x are deleted ﬁrst, then

w cannot be deleted by Lemma 1. This is a

contradiction as w ∈ D.

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234

If the two pixels x and w are deleted ﬁrst, then

q cannot be deleted by Lemma 1. This is a

contradiction as q ∈ D.

– Lastly, let us assume that the three pixels q, x,

and w are deleted at a time (i.e., in the same

subiteration). In this case

x is not SI-0-deletable, since it is not a 0-

border pixel (i.e., pixel q is black);

x is not SI-1-deletable, since it is not a 1-

border pixel (i.e., pixel p is black);

x is not SI-2-deletable, since it is not a 2-

border pixel (i.e., pixel w is black).

Thus all the three pixels q, x, and w need to be

SI-3-deletable. Consequently, pixels d, e, and

f are all white. It can be readily seen, by The-

orem 1, pixel q is simple iff all the three pixels

a, b, and c are also white. In this case, q is an

endpixel of type 2. Thus q is not SI-3-deletable,

and we arrive at a contradiction. (Similarly, by

Theorem 1, pixel w is simple iff all the three

pixels g, h, and i are also white. Since w is

an endpixel of type 2, it is not SI-3-deletable.

Thus we arrive at a contradiction.)

Since a non-SI-0-deletable border pixel may not

be SI-i-deletable (i = 0,1, 2,3) in the remaining

subiterations of the thinning process, this theorem

holds.

4.2 Examining Algorithm SF

Similarly to algorithm SI, we can state that algorithm

SF fulﬁlls the 1-attempt property:

Theorem 5. Algorithm SF is 1-attempt.

Proof. Without loss of generality, we can assume that

a border pixel p to be examined is in S

We need to verify that if p ∈ B in picture

,8, 4,B) is non-SF-0-deletable in the actual subit-

eration, it remains non-SF-0-deletable.

Let us assume indirectly that p is SF-0-deletable

for B\ D, where D ⊆ B\ {p} is a set of deleted pixels.

Since pixel p is a non-SF-0-deletable border pixel

for B, at least one of the last two conditions of Deﬁni-

tion 4 is violated. Consequently,

• p is an endpixel of type 1 for B, or

• p is non-simple for B.

It is obvious that if p is an endpixel of type 1 for

B (i.e., the ﬁrst point holds), p remains an endpixel of

type 1 for B \ D for any D ⊆ B\ {p}. Thus only the

second point is to be examined (i.e., p is non-simple

for B).

According to our assumption, p is deleted in a re-

maining thinning phase, thus it becomes simple for

B \ D. It means that N(p) ∩ D is a simpliﬁer set of

deleted pixels.

Since p is a non-simple pixel for B, at least one

template shown in Fig. 3 matches it. By Proposition

7 and Proposition 9 we can ignore the ﬁve templates

, T

, and T

. Thus T

is the only template to

be investigated.

If p is matched by T

, by Proposition 8, ex-

actly one of the two sets of pixels {q,w,x} ∩ B and

{s,t,u} ∩ B is a subset of all possible simpliﬁer sets

associated to p. Consequently,one of {q, w,x}∩B and

{s,t,u} ∩ B is to be completely deleted. Without loss

of generality, we can assume that ({q,w, x} ∩ B) ⊆ D.

It is known that black pixel x ∈ D (see Fig. 3), thus

the following four cases depicted in Fig. 10 are to be

checked:

• q 6∈ D,w 6∈ D (see Fig. 10a):

By Lemma 2, pixel x cannot be deleted. This is a

contradiction as x ∈ D.

• q 6∈ D,w ∈ D (see Fig. 10b):

By Proposition 2, x and w belong to different sub-

ﬁelds. Thus these two pixels cannot be deleted in

the same subiteration.

– If x is deleted ﬁrst, w cannot be deleted by

Lemma 1. This is a contradiction as w ∈ D.

– If w is deleted ﬁrst, x cannot be deleted by

Lemma 2. This is a contradiction as x ∈ D.

• q ∈ D,w 6∈ D (see Fig. 10c):

Similarly to the previous case, by Proposition 2, x

and q belong to different subﬁelds. Thus these two

pixels cannot be deleted in the same subiteration.

– If x is deleted ﬁrst, q cannot be deleted by

Lemma 1. This is a contradiction as q ∈ D.

– If q is deleted ﬁrst, x cannot be deleted by

Lemma 2. This is a contradiction as x ∈ D.

• q ∈ D,w ∈ D (see Fig. 10d):

By Proposition 2, x and q are in different sub-

ﬁelds, x and w are also in different subﬁelds.

(Note that q and w are in the same subﬁeld, see

Fig. 1b.) Then the following points are to be

checked:

– If just one of the two pixels q and w is deleted

ﬁrst, then we get the previously examined cases

shown in Fig. 10b and Fig. 10c, respectively.

Thus we arrive at a contradiction.

– If x is deleted ﬁrst, then both pixels q and w

cannot be deleted by Lemma 1. This is a con-

tradiction as q ∈ D and w ∈ D.

Since a non-SF-0-deletable border pixel in S

may

not be SF-i-deletable (i = 0,1,2, 3) in the remaining

subiterations, the proof by contradiction is completed.

1-Attempt 4-Cycle Parallel Thinning Algorithms

235

Table 1: Speed up C/A for the ﬁve test images shown in

Fig. 4-8 concerning the two 1-attempt algorithms, where

C is the computation time (in sec.) at the ‘conventional’

scheme, and A is the required runing time according to the

advanced ‘1-attempt’ implementation.

Test image

Speed-up

algorithm SI algorithm SF

1.305

0.046

= 28.37

0.468

0.037

= 12.65

0.720

0.036

= 20.00

0.264

0.032

= 8.25

0.182

0.011

= 16.55

0.067

0.010

= 6.70

0.374

0.038

= 9.84

0.166

0.037

= 4.49

0.403

0.067

= 6.01

0.208

0.061

= 3.41

5 USEFULNESS OF THE

1-ATTEMPT PROPERTY

A general and computationally efﬁcient implementa-

tion scheme for parallel thinning algorithms was pro-

posed in (Pal´agyi, 2008). This ‘conventional’ scheme

takes advantage of the fact that all thinning algorithms

may delete only border pixels. Thus we do not have

to examine the deletability of interior pixels, and the

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

tual ﬁnite picture) can be avoided by using a linked

list that stores the border pixels that are present in the

input picture of the actual thinning phase.

In our previous papers, we proposed ‘advanced’

implementation for 1-attempt thinning (Pal´agyi and

N´emeth, 2021a; Pal´agyi and N´emeth, 2021b), and we

gained remarkable speed up over the ‘conventional’

scheme.

The usefulness of the 1-attempt property of the

studied two 4-cycle algorithms SI and SF is demon-

strated by Table 1.

Both implementation schemes under comparison

were run on a usual laptop (Asus VivoBook S14; 1.8

GHz Intel Core i7; Fedora 32 Cinnamon, 64-bit) and

written in C++. Note that just the iterative thinning

process itself was considered here; reading the input

image and writing the output image were not taken

into account.

We strongly emphasize that the attainable speed-

up is quite image-dependent. Notice that algorithm SI

Figure 11: Comparison of the behavior of the ‘conven-

tional’ and the ‘1-attempt’ implementations of algorithm

SI for the image of a spider (see Fig. 6). While the ‘con-

ventional’ scheme evaluates the same border pixel several

times, the ‘1-attempt’ approach investigates each pixel just

once.

results in a larger speed-up than the achievable speed-

up of algorithm SF. This is why a border pixel my

be a ‘morefold’ border pixel but each pixel is in just

one subﬁeld. Figure 11 illustrates the differences be-

tween the ‘conventional’ and the ‘1-attempt’ imple-

mentations.

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