Detection of Local Symmetry Polylines of Polygons Based on Sweeping

Paradigm

Martin Safko

1,3 a

, Luka Luka

2 b

, Borut

Zalik

2 c

and Ivana Kolingerov

3 d

Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Praha 2, Czech Republic

Faculty of Electrical Engineering and Computer Science, University of Maribor,

Koro

ska cesta 46, SI-2000 Maribor, Slovenia

Department of Computer Science and Engineering, University of West Bohemia,

Technick

a 8, 306 14 Plze

n, Czech Republic

Keywords:

Symmetry, Polygon, Sweeping, Segmentation, Characterization, Computer Graphics.

Abstract:

Symmetry is a fundamental property of many objects of interest allowing us to simplify computation or re-

duce complexity and is also a contributing factor of beauty for a human brain. In this work, we try to identify

polylines satisfying the local reﬂection symmetry of polygons and show how to use them for shape character-

ization, segmentation or to ﬁnd global approximate reﬂection symmetries. We describe an algorithm based on

a sweep-line paradigm to efﬁciently compute the polylines by scanning through a polygon at various angles.

1 INTRODUCTION

Symmetry is one of the crucial features of geomet-

ric shapes, indicating their invariance to various ge-

ometric transformations or their combinations (Sun

and Bhanu, 2011). People perceive symmetrical pat-

terns as more attractive than their asymmetrical coun-

terparts (Enquist and Arak, 1994). Symmetry repre-

sents one of the main visible properties of objects and

is easily recognizable to the human eye. Unlike hu-

mans, the detection of symmetries is a considerably

challenging task for computers (

Zalik et al., 2022).

As the identiﬁcation of symmetries in shapes can sig-

niﬁcantly simplify further tasks of shape processing,

such as polygon segmentation or shape characteriza-

tion, numerous approaches for symmetry detection in

shapes have been proposed thus far.

Existing approaches for the detection of sym-

metries in shapes are often complex and resource-

intensive. Therefore, in this paper, we propose a

fast and simple method that extracts local features of

a shape using the concepts of sweeping and reﬂec-

tion symmetry. The shape, described by polygons, is

swept with sweep-lines at different angles. During

https://orcid.org/0009-0005-9403-8185

https://orcid.org/0000-0003-4691-5401

https://orcid.org/0000-0003-4372-5020

https://orcid.org/0000-0003-4556-2771

this procedure, polylines are obtained, which char-

acterize symmetrical areas of the shape. After the

sweeping part of the method, the polylines are ﬁl-

tered and combined to obtain a feature vector of the

shape. Due to the design of the method, the trans-

formations of input shapes do not inﬂuence the detec-

tion of local features. Furthermore, smaller degrees of

noise, present in a shape, do not signiﬁcantly impact

the sweeping process.

The remainder of this paper is structured as fol-

lows. In Section 2, previous work from the ﬁelds

of symmetry detection, polygon segmentation, shape

characterization, and sweeping are summarised. Sec-

tion 3 describes the proposed method and its appli-

cations. In Section 4, results of the local feature de-

tection method are presented. Section 5 discusses the

future work. The paper is concluded in Section 6.

2 RELATED WORK

This section consists of four parts. In the ﬁrst part, the

sweeping technique is explained shortly. The symme-

try detection methods are described and brieﬂy sur-

veyed in the second part. After that, methods for poly-

gon segmentation are discussed. Lastly, shape charac-

terization methods are described.

Safko, M., Luka

c, L., Žalik, B. and Kolingerová, I.

Detection of Local Symmetry Polylines of Polygons Based on Sweeping Paradigm.

DOI: 10.5220/0012473400003660

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 19th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2024) - Volume 1: GRAPP, HUCAPP

and IVAPP, pages 329-336

ISBN: 978-989-758-679-8; ISSN: 2184-4321

329

2.1 Sweeping

Sweeping is a well-established technique for efﬁ-

cient solutions to geometric problems (Preparata and

Shamos, 1985). The basic element of sweeping in

2D is a sweep-line, which moves through a geomet-

ric space and stops at event points, speciﬁc to various

geometric problems. A local part of the problem is

solved during the stop. All local solutions are stored

in a data structure called sweep-line status. When all

event points have been reached by the sweep-line, the

geometric problem is completely solved. The same

approach is utilized in 3D, where a sweep-plane is

used for sweeping instead of a sweep-line.

The sweeping concept has been used in many dif-

ferent ﬁelds, such as the construction of a Delau-

nay triangulation (

Zalik, 2005; Domiter and

Zalik,

2008), the construction of Voronoi diagrams (For-

tune, 1986; Jin et al., 2006), convex hull computa-

tion (Borna, 2019), state space exploration (Chris-

tensen et al., 2001; Jensen et al., 2012), spatial clus-

tering (

Zalik and

Zalik, 2009), and image segmenta-

tion (Wu et al., 2014).

2.2 Symmetry Detection

In 2D, there are four basic symmetry types: reﬂection,

rotational, translational, and glide-reﬂection (Liu

et al., 2010). In the continuation, only reﬂection sym-

metry is considered. Symmetry can be either global

or local (also referred to as partial). Global symmetry

indicates that the whole shape is symmetric while lo-

cal symmetry signiﬁes that the shape contains smaller

symmetrical segments.

There are plenty of methods, dealing with the de-

tection of global symmetries. Various techniques are

used, such as: ﬁnding symmetry axes from potential

ﬁelds (Prasad and Yegnanarayana, 2004), building ro-

bust structure descriptors (Atadjanov and Lee, 2016),

grid splitting (

Zalik et al., 2022), and applying Hough

transforms (Lei and Wong, 1999).

Local symmetry detection is even more challeng-

ing. Still, many methods were proposed: correlation-

based (Masuda et al., 1993), voxelisation-based

(Podgorelec et al., 2023), approach using surface

descriptors (Gal and Cohen-Or, 2006), neural net-

works (Tsogkas and Kokkinos, 2012), and other.

2.3 Polygon Segmentation

The procedure for partitioning a complex polygon

into simpler polygons is called polygon segmenta-

tion. In the past, the main motivation for this task

was speed improvement of various triangulation algo-

rithms (O’Rourke, 1998). Despite that, the segmenta-

tion of polygons often represents a standalone task.

There are several approaches to polygon segmenta-

tion: monotone partitioning (Lee and Preparata, 1977;

Wei et al., 2012), triangulation (Chazelle, 1991),

trapezoidation (Hain and Langan, 2005), and division

into convex polygons (Fern

andez et al., 2000).

2.4 Shape Characterization

One of the main topics in the ﬁeld of image process-

ing is the characterization of geometric shapes. The

main idea behind this procedure is to detect important

features in a shape, extract them, and store them in

a feature vector (Zhang and Lu, 2004). Such shape

representations can be used for various tasks, e.g.

classiﬁcation of a shape, detection of similar shapes,

and data preprocessing in machine learning. Unfor-

tunately, the selection of appropriate features is not

a trivial task. Numerous approaches were proposed

to solve this issue. However, many of the existing

methods possess some weaknesses, such as sensitiv-

ity to noise (Blum, 1967), non-unique characteriza-

tion of equal shapes with different transformations

(e.g. translation, rotation, scaling) (Grosky et al.,

1992), and inability to successfully process shapes

with holes (Iivarinen and Visa, 1996).

Generally, characterization methods can be classi-

ﬁed into two larger groups (Loncaric, 1998). Exter-

nal methods process the shape boundary and perform

the characterization upon boundary points. Internal

methods extract features from the shape interior.

External methods use various approaches for

shape characterization: boundary representation with

1D function (Bennett and Mac Donald, 1975; Wang

et al., 1994), Fourier transform of the boundary (Zahn

and Roskies, 1972; Pinkowski, 1993), stochastic al-

gorithms (Das et al., 1990), detection of critical

points in chain codes (Freeman, 1978), scale-space

representation (Witkin, 1987), and many others. Most

common internal characterisation methods are medial

axis transform (Blum, 1967; Peleg and Rosenfeld,

1981; Makem et al., 2020), shape decomposition into

simpler shapes (Bjorklund and Pavlidis, 1981; Attene

et al., 2009), and detection of shape features using

sweep-line (

Zalik et al., 2023).

Nowadays, shape characterization methods often

rely on deep neural networks. They are used in agri-

culture (Toda et al., 2020), medicine (Xu et al., 2020;

Hasan et al., 2022), physics (Bisheh et al., 2023), ar-

chitecture (Yan et al., 2019), and many others. De-

spite yielding great characterization accuracy, the dis-

advantage of those methods is the need for huge

datasets during the training phase of their models.

GRAPP 2024 - 19th International Conference on Computer Graphics Theory and Applications

330

3 SWEEP-LINE METHOD

Let P = {p

,..., p

}, p

∈ R

be a polygon represent-

ing the input shape in the Euclidean plane. Our goal

is to ﬁnd a suitable set of polylines, also referred to

as chains, {M

,..., M

} that in some sense character-

ize local symmetry of the input polygon and can be

used to efﬁciently compute geometric properties such

as generalized reﬂection symmetry, polygon segmen-

tation or shape characterization, described later.

The algorithm works, as the name suggests, by

sweeping a line over the polygon several times with

increasing angle. The resulting polylines from each

run are then ﬁltered and combined to produce the ﬁnal

solution. We describe each part in more detail in the

following text. The main structure of the algorithm is

shown in Alg. 1.

Function generatePolygonChains

Input : polygon P

Output: set of chains from all angles C

C ← ∅;

angle ← 0;

while angle ≤ 2π do

polygon ← rotate(angle,P);

chains ← f indChains(polygon);

C ← C ∪ chains;

angle ← increment(angle);

end

C ← f ilter(C);

C ← combine(C);

return C;

end

Function ﬁndChains

Input : polygon P = {p

,.. . , p

}

Output: chains = {M

,.. . ,M

}

chains ← ∅;

sortedPoints ← sort(points(P));

for k = 1, . .., n do

← sortedPoints[k];

k−1

, p

k+1

← neighbors(p

, P);

M ← selectChain(chains, p

k−1

, p

k+1

);

int

← intersect(P, p

);

← (p

+ p

int

)/2;

M ← M ∪ m

;

assignChain(M, p

);

end

return chains;

end

Algorithm 1: Sweep line symmetry pseudocode.

3.1 Sweeping Procedure

In this section, we will assume a horizontal sweep line

since sweeping a line at an angle over a polygon is

equivalent to sweeping a horizontal line over a rotated

polygon. The f indChains procedure starts by sorting

the input vertices by their y coordinate. The sweep

line starts at the bottom-most vertex of the polygon.

If such vertices are multiple we pick the middle point.

Next, we move the line upward, vertex by vertex, pro-

gressively building local symmetry polylines until we

reach the last vertex.

int

k+1

k-1

i-1

i-2

Figure 1: One step of the sweeping procedure.

Consider a sweep line s passing through point p

(see Fig. 1), where m

i−1

and m

i−2

are symmetry poly-

line points from the previous iterations. Let us deﬁne

e as the closest edge of the polygon intersecting with

the sweep line s such that the resulting line segment is

inside P, and deﬁne p

int

as the intersection point. We

extend the symmetry polyline {.. .,m

i−2

i−1

} with

the midpoint

int

+ p

and continue on the next iteration.

3.1.1 Polyline Selection

At each iteration, a chain M is selected ac-

cording to the neighboring polygon vertices

(selectChain(chains, p

k−1

, p

k+1

). Since every

vertex can be associated with only one chain, we can

store the chain index with each point. It is important

to notice that there are cases where several chains

can be selected for extension as can be seen in Fig 2.

There are three cases to consider:

1. no neighboring vertices have been processed yet

Detection of Local Symmetry Polylines of Polygons Based on Sweeping Paradigm

331

2. one of the neighbors has been processed

3. both neighboring vertices have been processed

In the ﬁrst case, we create a new chain starting at

the current vertex. In the second case, we extend the

chain belonging to the neighboring vertex. In the third

case, there are multiple choices to consider. We could

either extend both chains, select just one according to

some criterion, or create a new chain. We decided on

extending both chains, which gave us the best results.

Figure 2: Both chains being extended (left) or a completely

new chain created (right).

3.1.2 External Chains

A small modiﬁcation of the polygon intersection rou-

tine yields chains outside of the polygon (Fig. 3)

and can provide additional information about the in-

put shape, which is useful in some applications.

The intersect(P, p

) function is changed to choose

a line segment outside the polygon if it exists. More

speciﬁcally, we pick the closest edge e such that the

resulting line segment L satisﬁes L ∩ P = ∅. This is

equivalent to running the unmodiﬁed algorithm on the

complement of an input polygon.

Figure 3: Left: internal chains Right: external chains.

3.2 Chain Filtering and Combining

After all the chains have been created for each sweep

angle, they are further processed to produce the ﬁnal

symmetry polylines. This processing includes ﬁlter-

ing, combining and separating chains to suit the needs

of subsequent use cases, see examples in Fig 4.

First, short chains with only a few points that

resulted from small convex/concave irregularities on

the polygon boundary are discarded. The threshold

for the number of discarded points can be adjusted

based on a desired sensitivity to noise. Next, chains

with long jumps between successive points are split

into individual chains. What is considered too long

depends on the distance between points before the

jump. Finally, if enabled, chains are combined to

form larger structures, useful namely to ﬁnd global

properties such as generalized reﬂection symmetry,

described next.

3.3 Generalized Reﬂection Symmetry

We deﬁne symmetry as any transformation T (X) such

that T (X) = X, i.e. applying the transformation yields

the same object. However, in the real world, no ob-

ject is perfectly symmetric, so we replace the strict

equality sign with approximate equality.

Reﬂection or mirror symmetry is deﬁned over a

straight line l : n · p − d = 0, where n is the normal

vector and d is the distance from the origin to the line.

The resulting transformation can then be deﬁned as

r(p

, l) := p

− 2(n · (p

− dn))n

where p

is a point we want to reﬂect. If the condition

r(p

, l) = p

holds for all points of a polygon then

we say it has a reﬂection symmetry. Since we are not

interested in perfect but in approximate symmetry, we

can rewrite it as an optimization problem where we

try to ﬁnd a line that minimizes the residual, namely

min

∑

i̸= j

∥ r(p

, l) − p

∥

We generalize this deﬁnition by replacing the

straight line with a curve. This introduces the prob-

lem of having many different curves with wild shapes

satisfying the symmetry condition. So, we add a loss

function for the shape of a curve with a straight line

being a minimum. The ideal function satisfying these

properties is a curvature. Therefore, we can specify

the solution to generalized reﬂection symmetry as an

optimization problem

min

κ(s)ds

where κ(s) is the curvature at point s and the curve C

satisﬁes the reﬂection condition. We can combine all

conditions into one loss function such as

min

∑

i̸= j

∥ r(p

, C) − p

∥ +α

κ(s)ds

where α ≥ 0 is a user-deﬁned parameter controlling

the straightness of the reﬂection curve.

Symmetry chains enable us to easily ﬁnd suitable

solutions to the generalized symmetry of a polygon.

First, chains are combined based on their distance and

tangent directions of endpoints. These larger pieces

GRAPP 2024 - 19th International Conference on Computer Graphics Theory and Applications

332

Figure 4: Example of different objects with their symmetry chains.

Figure 5: Individual chains (left) and approximate general-

ized reﬂection symmetry (right).

then have curvature reduced by smoothing to decrease

the general symmetry loss function. This process is

repeated until the loss improvement stops and so the

result is only a local optimum. Still, the results for our

data are satisfactory, see Fig.5.

3.4 Polygon Segmentation

Polygon segmentation is the process of partitioning

a polygon into individual parts with geometric sig-

niﬁcance. We use the symmetry chains computed

by our algorithm to ﬁnd regions of interest speciﬁed

by custom ﬁlter rules. These rules describe proper-

ties used to select chains associated with desired seg-

ments, e.g. min/max length, width of the segments,

or chain sweep line angle. Consider a shape in Fig.

6. Choosing long chains with small widths yields the

ﬁngers part of the hand. Additionally, a constraint on

chain angle allows us to select any of the ﬁve ﬁngers.

Similarly, picking wide chains results in the selection

of the palm. Incorporating external chains even al-

lows us to specify empty space between solid parts.

3.5 Shape Characterization

The input polygon is moved so that its center of mass

coincides with the origin and is scaled to range [−1, 1]

to ensure output chain invariance to scale and transla-

tion, which is necessary for a practical shape charac-

Figure 6: A shape with several narrow sections at different

orientations and a wide section at the base.

terization procedure. We follow the steps of (

Zalik

et al., 2023) and implement similar length-based fea-

tures. Additionally, we include the total curvature of

a chain to improve the original approach. The feature

vector of the object i is therefore deﬁned as



f (M

), g(M

)



, ...,



f (M

), g(M

)



where f (M) is the length and g(M) is the total curva-

ture of the chain M. Two objects i and j are consid-

ered equal when the two feature vectors V

and V

are

compatible within a reasonable margin of error, i.e.

| = |V

| and V

≈ V

for all indices k.

External chains can be used to augment the exist-

ing features, or separately, when the input shape con-

tains self-intersections or other imperfections that af-

fect the internal structure of the polygon.

4 RESULTS

We present several examples that showcase our algo-

rithm. Fig. 7 compares simple shapes with more com-

plex ones. Notice how the front wheel on the bicycle

produces the same chain as in the simple circle. In

Fig. 8 we show examples of generalized symmetry

curves. In Fig. 9 we demonstrate the robustness of

Detection of Local Symmetry Polylines of Polygons Based on Sweeping Paradigm

333

Figure 9: Invariance of local symmetry chains to small changes in boundary. Top: boundary noise is increased from left to

right. Bottom: boundary is smoothed out from left to right.

Figure 7: A selection of chains shown with various shape

complexities. Chains are split at places where it would re-

sult in a long jump.

local symmetry chains to the noise of polygon bound-

ary. In the ﬁrst row, the boundary gets progressively

noisier with minimal distortion to chains. Similarly,

the second row shows a smoothing effect. Again,

the chains are mostly intact, which illustrates the re-

silience of the algorithm in poor conditions, which is

useful especially for shape characterization.

All images were generated and exported as SVG

Figure 8: Generalized symmetry curves computed on se-

lected objects.

ﬁles by our algorithm. Input data was taken from pre-

vious work done by (

Zalik et al., 2023). Original input

data was converted from chain codes to a list of ﬂoat-

ing point vertex coordinates to make the actual com-

putation easier. This freed us from having to work on

a pixel lattice and allowed processing of an arbitrary

rotated input shape.

We evaluate our method on different shapes and

compare time performance with other method in Ta-

ble 1. The experiments were done on a laptop with an

Intel i7-8550U CPU @ 1.80GHz and 8GB RAM.

5 FUTURE WORK

The algorithm can be further extended to process

smooth shapes with boundaries represented by curves

and to produce chains also represented as curves

which might have other interesting applications.

Other possible ventures might be local deformation

GRAPP 2024 - 19th International Conference on Computer Graphics Theory and Applications

334

Table 1: Performance comparison with (

Zalik et al., 2023).

Shape #points

Zalik et.al. [ms] Ours [ms]

Bird 2372 693 95

Dolphin 2870 1098 136

Hand 3798 936 170

Buddha 10146 24693 941

Ballet 14438 15590 1737

Cupid 20646 35302 3232

Spider 23900 36513 4112

of the input shape controlled by the symmetry chains,

which could be helpful for animating polygons.

6 CONCLUSION

We presented an algorithm that extracts symmetry

chains of a polygonal shape by utilizing the local re-

ﬂection symmetry. The algorithm is based on the

sweep-line paradigm for efﬁcient processing of the

polygon. Due to its generality, resulting local sym-

metry chains can be used for polygon segmentation,

shape characterization and ﬁnding generalized reﬂec-

tion symmetry. We demonstrated resilience to local

changes in boundary due to noise or smoothing, and

described possible future applications.

ACKNOWLEDGEMENTS

This research was supported by the Czech Sci-

ence Foundation under research project 21-08009K,

the Slovene Research and Innovation Agency un-

der research project N2-0181, Research Programme

P2-0041, and the Charles University grant SVV

260699/2023.

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