COMPUTATIONAL SYMMETRY VIA PROTOTYPE DISTANCES

FOR SYMMETRY GROUPS CLASSIFICATION

M. Agust

ı-Melchor,

Angel Rodas-Jord

a and J. M. Valiente-Gonz

alez

Computer Vision Group, DISCA, ETS. Ingenieria Inform

atica

Universitat Polit

ecnica de Val

encia (UPV), Camino de Vera, s/n, 46022 Valencia, Spain

Keywords:

Computational symmetry, Symmetry groups, Prototype-based classiﬁcation, Adaptive nearest neighbour clas-

siﬁcation.

Abstract:

Symmetry is an abstract concept that is easily noticed by humans and as a result designers make new creations

based on its use, e.g. textile and tiles. Images of these designs belong to a more general group called wallpaper

images, and these images exhibit a repetitive pattern on a 2D space. In this paper, we present a novel compu-

tational framework for the automatic classiﬁcation into symmetry groups of images with repetitive patterns.

The existing methods in the literature, based on rules and trees, have several drawbacks because of the use of

thresholds and heuristics. Also, there is no way to give some measurement of the classiﬁcation goodness-of-ﬁt.

As a consequence, these methods have shown low classiﬁcation values when images exhibit imperfections due

to the manufacturing process or hand made process. To deal with these problems, we propose a classiﬁcation

method that can obtain an automatic parameter estimation for symmetry analysis. Using this approach, the

image classiﬁcation is redeﬁned as distance computation to the binary prototypes of a set of deﬁned classes.

Our experimental results improve the state of the art in symmetry group classiﬁcation methods.

1 INTRODUCTION

Symmetry is an abstract concept that is easily noticed

by humans and as a result designers make new cre-

ations based on its use. In industrial sectors, like tex-

tile, ceramics or graphic arts, the notion of symmetry

is always present as an aesthetic element, indispens-

able in every new design. When using a symmetrical

design, a pattern is repeated to ﬁll the workplane, fol-

lowing strict placement rules. The traditional Tillings

and Patterns Theory describes the fundamentals of

this design creation process. The results are designs

such as those shown in Figure 1. These are images of

typical wovens samples and mosaics samples, com-

monly referred as regular mosaics, wallpaper images,

wallpaper patterns, or simply Wallpapers. However,

little effort has been made in the area of wallpaper

image analysis and classiﬁcation, and so this work ex-

plores this direction.

Tile and textile companies have thousands of these

samples accumulated over the years and stored in

company storerooms or museums. In most cases,

these collections are digitized and stored in images

databases. These collections are an invaluable asset

used as source of inspiration for designers. But de-

signers suffer serious limitations when searching and

manage these information because the images are in

bitmap format, and designers are accustomed to use

other abstract terms related with perceptual criteria

(Closeness, Co-circularity, Co-linearity, Overlapping,

etc..). Therefore some kind of image analysis is nec-

essary to extract information about the internal geom-

etry and structure of these patterns. All this informa-

tion, conveniently organized, can be used to build an

Information System that allows all these historical de-

signs to be effectively used. The compact image de-

scription obtained can also be used in other areas such

as object detection or Content Based Image Retrieval.

The term computational symmetry refers to the al-

gorithmic treatment of symmetries. The recent resur-

gance interests in computational symmetry, for com-

puter vision and computer graphics applications, has

been recognized by the recent tutorial at ECCV 2010

Conference (Liu et al., 2010) that explored the funda-

mental relevance and potential power that computa-

tional symmetry enables. But, as is stated by these

authors, using real world data: it ”turns out to be

challenging enough that, after decades of effort, a

fully automated symmetry system remains elusive for

real world applications”. The authors also summa-

Agustí-Melchor M., Rodas-Jordá Á. and M. Valiente-González J..

COMPUTATIONAL SYMMETRY VIA PROTOTYPE DISTANCES FOR SYMMETRY GROUPS CLASSIFICATION.

DOI: 10.5220/0003375300850093

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2011), pages 85-93

ISBN: 978-989-8425-47-8

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Details of wallpaper images obtained from (Wikipedia, ), (Edwards, 2009), and (Joyce, 2007) collections. They

are images of real textile samples, showing hand-made artefacts and damaged or scratched parts. The geometry of the Unit

Lattice is draw on the centre of each image.

rize two key points. Firstly, to compute the concepts

of group theory from imperfect, noisy, ambiguous,

distorted and often hidden symmetry patterns in real

world data. And secondly, to apply the mathematical

theory of symmetry, group theory, to algorithms that

incorporate/assume the limitations of representational

power of computers.

In this paper, in order to explicitly addressing each

of these statements, we propose a novel computa-

tional framework based on continuous measures of

symmetries to classify real and synthetic images of

two-dimensional repetitive patterns. These will be

formally described as symmetry groups (wallpaper

groups or plane symmetry groups or plane crystallo-

graphic groups), based on the symmetries in the pat-

tern. Some examples of textile and tile patterns are

shown in Figure 1.

A symmetry of any 2D pattern can be described

through a set of geometrical transformations that

transforms it in itself. Those transformations that pre-

serve distances are know as isometries, and the only

plane isometries are: translations, rotations (n-fold)

reﬂections (specular) and glide reﬂections (specular

plus lateral displacement). Therefore, in a pattern up

to four different kinds of symmetries can be observed,

one external to the pattern (translational symmetry),

and the other three internals (rotation, reﬂection and

glide reﬂection symmetries).

Regarding the translational symmetry, a wallpa-

per pattern is a repetition of a parallelogram shaped

subimage, called Unit Lattice (UL), so that the full

pattern can be reconstructed by the replication (dis-

placements) of this UL. A lattice extraction proce-

dure is then needed to obtain the lattice geometry, in

the form of two direction vectors (UL sides) that de-

ﬁne the translational symmetry of the pattern. Among

others, autocorrelation ((Liu and Collins, 2000),(Liu

et al., 2004)), Fourier or wavelet (Agust

ı et al., 2008)

approaches have been used, among others, to solve

this question. In this work we assume that the lat-

tice geometry has been already obtained and the UL

is known.

For internal symmetries, we rely on the Symme-

try Groups theory (Horne, 2000), which uses the con-

cept of symmetry groups to formulate a mathematical

description of complex plane pattern structures. The

set of isometric transformations that brings a ﬁgure

in coincidence with itself is called a symmetry group.

In the 1-D case, Frieze Symmetry Groups (FSG), the

pattern contains one traslational symmetry and some

other isometries. In the 2D case, Plane Symmetry

Groups (PSG), the pattern contains two traslational

symmetries (lattice) and some other isometries. For

example, the pattern in Figure 1 (left) has only trans-

lational symmetry. In contrasts, the other patterns of

Figure 1 have more isometries, such as 180

◦

rotations

and reﬂections about two axes, and the last pattern can

be reconstructed using 120

◦

rotations.

It is also known that there is a geometric restric-

tion, called the ’crystallographic constraint’, which

limits the number of possible rotations that can be

applied to a motif or tile to completely ﬁll the plane

without holes nor overlapping. These rotations are

180

◦

,120

◦

,90

◦

and 60

◦

. Accordingly, the number of

PSG are limited to 17 cases, which is one of the main

issues that help us to describe the pattern structure.

Figure 2 shows the details of each PSG, as well as

their standard notation. For example, the patterns

in Figure 1 belong, from left to right, to symmetry

groups P1, PMM, PGG and P31M.

2 STATE OF THE ART

An extended discussion and comparison of Symme-

try Detection algorithms can be found in (Liu et al.,

2010). The authors have choosen three algorithms be-

cause they are ”(1) recently published; (2) have gone

beyond single reﬂection/rotation symmetry detection.

In fact, they all claim to detect multiple symmetries

in an image; (3) directly applicable to un-segmented,

real images; and (4) the authors make their source

VISAPP 2011 - International Conference on Computer Vision Theory and Applications

Figure 2: Visual representation of the 17 Wallpaper Groups including their standard notation, the reﬂection symmetry axes

(glide and non-glide) and rotation centres of n-fold symmetries. The UL is referred as Fundamental Parallelogram.

code publicly available”. They use a test bed of 176

images with hand-labeled ground truth from image

databases of object-detection challenges, divided ac-

cording to (i) synthetic versus real images; and (ii)

images containing a single dominant symmetry ver-

sus multiple symmetries, considering only two types

of symmetries (rotation and reﬂection). To quantify

algorithms performance, a success rate over the test

image dataset is computed obtaining a 63% of best

mean sensitivity for reﬂection over all types (synthetic

or real, single or multiple symmetries) of images. In

real images, the best mean sensitivity rate is 42% (for

reﬂection). However, for a true understanding of mul-

tiple existing symmetries in an image, we should be

more concerned with the overall success rate S1 (in-

cluding the false positives). The best values of S1

are much worse: 27% for reﬂection/rotation symme-

try detection in the overall image set, and 19% on real

images.

On the other side, the classical algorithms to cata-

logue wallpaper designs are procedures to be used by

humans based on heuristics ((Grunba

um and Shep-

ard, 1987), (Schattschneider, 1978)). They are pro-

posed in the form of decision trees whose branches

are raised as questions that humans can answer by

looking at the design. The ambiguities in the rules

formulation poses complexity into the computer re-

formulation of these rules as computer programs.

The main computer autonomous approach of this

kind has been made by Liu et al. ((Liu and Collins,

2000), (Liu et al., 2004)) in which a computer model

is proposed for the automatic classiﬁcation of frieze

and plane groups on grayscale bitmap images. This

model expresses the Schattschneider algorithm in the

form of a rule-based classiﬁer (RBC) where each

symmetry group corresponds to a unique sequence of

yes/no answers to whether the pattern contains some

symmetries, such as n-fold rotation or reﬂections. It

can be seen as a kind of decision tree classiﬁer with

binary symmetry values. These values are computed

as scores as follows. Firstly, the maximum of the cor-

relation between the transformed image (by applying

the symmetry that has been measured) and the origi-

nal image is obtained. Secondly, at the position of this

maximum, an UL is extracted from the transformed

image, and the SSD (sum of squared differences) be-

tween corresponding intensity values of the original

and transformed UL is computed. As the classiﬁca-

tion needs binary values, a thresholding step is re-

quired. In a initial work (Liu and Collins, 2000),

an experimentally obtained ﬁxed threshold value was

used. In a later work (Liu et al., 2004), a most ro-

bust procedure was introduced using a χ

(N) test and

computing the threshold as a function of N.

Our experience conﬁrms that both methods can

be tuned to obtain good results for the Joyce (Joyce,

2007) dataset. Even so, the ﬁrst method of obtaining a

threshold cannot be generalized to other datasets be-

cause it has been experimentally obtained. The sec-

ond method assumes that translational symmetry has

non-zero standard deviation, so it can not be applied

in ideal synthetic images. In addition, the use of RBC

obtains only one classiﬁcation result without an asso-

ciate measure of conﬁdence. Therefore, it becomes

necessary to enhance this solution - as is also noted in

(Reddy and Liu, 2005).

COMPUTATIONAL SYMMETRY VIA PROTOTYPE DISTANCES FOR SYMMETRY GROUPS CLASSIFICATION

In this work, we propose the use of continuous

features for a distance-based classiﬁcation approach

and the use of binary class prototypes describing the

17 PSG mentioned above, and which are adaptively

adjusted for image conditions. In this way, the high

degree of variance of symmetry features, due to noise,

deformations or just the nature of the hand made prod-

ucts, is asssumed.

3 PROPOSED METHOD

3.1 Feature Computation

We started by using a nearest neighbour classiﬁer

(NNC) as this enabled us to obtain a measure of good-

ness for the classiﬁcation. This kind of method re-

quire the use of continuous-value feature vectors. A

close view of symmetry groups description in Figure

2 reveals that the minimum number of symmetry fea-

tures needed to distinguish every PSG is twelve. They

are four (R

, R

and R

) related to rotational sym-

metries (2-fold, 3-fold, 4-fold and 6-fold); four (T

, T

and T

) to describe reﬂection simetries (non-

glide and glide) along axes parallel to the sides of UL;

and four more features (D

, D

and D

) for re-

ﬂection (non-glide and glide) with respect to the two

UL diagonals.

We deﬁned a symmetry feature vector (SFV) of

twelve elements that identiﬁes the presence/absence

of these image symmetries as follows:

SFV = ( f

, f

, ..., f

); f

∈ ℜ, 0 ≤ f

≤ 1 (1)

We compute the twelve symmetry features f

, by

using a slightly different method to the Liu’s team.

To obtain a symmetry value for a speciﬁc isometry,

e.g 2-fold rotation, we apply this transformation to

the original image I(x, y) obtaining the transformed

image I

(x, y). A piece of the transformed image,

of the same size that the UL, is taken. A bound-

ing box of the UL is actually used BBox(x

, y

), to

save computations. A score map is then computed

as Map(x, y) = 1 − SAD(x, y), where SAD(x, y) is the

sum of absolute differences between the transformed

BBox and the original image at (x,y) position, and it

is scaled to [0, 1] by BBox size m, according to:

SAD(x, y) =

∑

I(x − x

, y − y

) − BBox(x

, y

)

If the symmetry is present in the image, this

map peaks at several positions indicating the pres-

ence of that symmetry, while revealing lower val-

ues in areas where the symmetry is not hold. The

maximum − minimum

difference should then be a

good measure to quantify that feature. Figure 3 shows

an example. However, there are patterns without in-

ternal symmetries, such as P1 patterns (see Fig 2)

so that max-min difference should be relative to any

other value representing the presence of symmetry.

The only symmetry always present in every pattern is

the traslational symmetry (S

). So we can obtain S

by computing the previous score map using the orig-

inal image and its translated version (following UL

sides) and the maximum of this map will represent the

upper score level of symmetry S

present in the im-

age. Finally, we compute the normalized components

of the Symmetry Feature Vector (SFV) as follows:

max(Map)−min(Map)

−min(Map)

1 ≤ i ≤ 12 (2)

The higher the value of f

, the more likely the image

contains symmetry. Table 1 shows the SFV vectors

obtained for the four wallpaper samples in Figure 1.

As expected, these results partially conﬁrm high val-

ues that indicate the presence of symmetry and low

values otherwise. For example, sample 2 is a PMM

wallpaper group, so it must have R

, T

and T

sym-

metries, and correspondingly its SFV shows values

higher than 0.80 for those features. The same hap-

pens in sample 3, with very high values (> 0.9) for R

G and T

G symmetries, which is expected in a PGG

wallpaper. However, sample 3 (P31M) have features

with values of 0.7 and 0.65 denoting the presence of

symmetry, while these values would be considered

as indicating non-symmetry in the other cases. On

the contrary, the ﬁrst sample is a P1 with no symme-

tries, but its SFV shows values up to 0.8, what would

be considered as symmetry in the other cases. Be-

cause these features were computed as gray level dif-

ferences between image patches, so their values will

strongly depend on the particular arrangements of im-

age pixels, i.e. the image complexity. As a conse-

quence SFV requires a higher level of adaptation to

the image conditions, i.e. taking into account on the

contents of each image separately. This idea will be

used later.

3.2 Symmetry Groups Classiﬁcation

To classify a wallpaper image, featured by SFV, we

need to learn or deﬁne a set of class samples. Fortu-

nately, the number of classes (17) and their structure

are known in advance. For the sake of simplicity, we

VISAPP 2011 - International Conference on Computer Vision Theory and Applications

Table 1: Symmetry feature vectors of the four wallpapers showed in Figure 1. The bold values means a value that has to be

considered as presence of symmetry to consider each vector as the group that it belongs to, while the others mean absence of

others symmetries.

Sample SFV=(R

, R

, T

, D

, T

, D

) PSG

1 (0.62, 0.47, 0.69, 0.34, 0.65, 0.67, 0.80, 0.59, 0.37, 0.43, 0.80, 0.59) P1

2 (0.82, 0.09, 0.20, 0.09, 0.88, 0.83, 0.20, 0.19, 0.27, 0.26, 0.2, 0.19) PMM

3 ( 0.95, 0.42, 0.33, 0.46, 0.39, 0.45, 0.31, 0.48, 0.98, 0.99, 0.31, 0.48) PGG

4 (0.46, 0.69, 0.28, 0.49, 0.74, 0.65, 0.48, 0.72, 0.74, 0.65, 0.48, 0.72) P31M

Table 2: Binary prototypes for the 17 PSG classes.

Classes Prototype

Feature vectors

P1 (0,0,0,0,0,0,0,0,0,0,0,0)

P2 (1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,1,0,0,0,1,0)

(0,0,0,0,0,0,0,1,0,0,0,0)

PMM (1,0,0,0,1,1,0,0,0,0,0,0)

PMG

(1,0,0,0,1,0,0,0,0,1,0,0)

PMG

(1,0,0,0,0,1,0,0,1,0,0,0)

PGG (1,0,0,0,0,0,0,0,1,1,0,0)

Classes Prototype

Feature vectors

CMM (1,0,0,0,0,0,1,1,0,0,1,1)

P4 (1,0,1,0,0,0,0,0,0,0,0,0)

P4M (1,0,1,0,1,1,1,1,0,0,1,0)

P4G (1,0,1,0,0,0,1,1,1,1,1,0)

P3 (0,1,0,0,0,0,0,0,0,0,0,0)

P31M

(0,1,0,0,1,1,1,0,1,1,1,0)

P31M

(0,1,0,0,1,1,0,1,1,1,0,0)

P3M1

(0,1,0,0,0,0,1,0,0,0,1,0)

P3M1

(0,1,0,0,0,0,0,1,0,0,0,0)

P6 (1,1,0,1,0,0,0,0,0,0,0,0)

P6M (1,1,0,1,1,1,1,1,1,1,1,0)

start by proposing the use of binary prototypes repre-

senting each one of the 17 classes. Each prototype has

a SFV with components to ’1’ if the symmetry is hold,

and ’0’ otherwise. The Table 2 shows the resulting

23 prototypes. Some classes have two prototypes be-

cause there are two possibilities where reﬂection sym-

metry can appears (two UL sides and two diagonals).

We then use a Nearest Neighbour Classiﬁer (NNC) to

perform the task. The Euclidean distance to the class

prototype can be used as a measure of conﬁdence.

After applying the NNC to several image collec-

tions we did not found signiﬁcant improvements in

comparision with RBC (see the Experiments section).

It is probably due to the bias of the feature values.

The minimum values (non-symmetry) are higher than

expected because the SAD between image patches al-

ways produces non-null values. Moreover, maximum

values (symmetry) rarely approximate to traslational

symmetry. In this situation, the use of binary proto-

types, with inter-class boundaries equidistant to each

class, does not ﬁt the problem. However, some ad-

vantage has been achieved. The NNC produces an or-

dered set of outputs describing the class membership

of each sample. This latter consideration can enable

an automatic adjustment of the prototypes in order to

adapt them to the image variability.

3.3 Adaptive NNC (ANNC)

Recent works on NN classiﬁers have shown that adap-

tive schemes (Wang et al., 2007) outperforms the re-

Figure 3: Original image (up) width the unit lattice (UL)

and score map (down) for 2-fold symmetry rotation.

sults of classic NNC in many applications. We adopt

this approach for several reasons. Firstly, the discrete

nature of the image domain introduces errors in the

computations of image transforms. However, when

COMPUTATIONAL SYMMETRY VIA PROTOTYPE DISTANCES FOR SYMMETRY GROUPS CLASSIFICATION

assuming perfect symmetries (synthetic images), the

absence of symmetry cannot be computed as ’0’ nor

can perfect symmetry be computed as ’1’. Secondly,

the dynamic range of the symmetry values ( f

) is ex-

tremely variable, depending on the speciﬁc conditions

of each image. In simple images (e.g. line-drawings)

the presence of large homogeneous areas in the image

leads to high symmetry scores, even in the absence of

symmetry. In some images, the presence/absence of

certain symmetries is due to small details, leading to

symmetry components being compressed into a nar-

row range. In other cases, symmetries are diminished

by a particular symmetry score that is stressed by spe-

ciﬁc pixel arrangements in the image. A further con-

cern is that the null-class P1 (total absence of interior

symmetries) is the main source of false positives.

In response to this situation we propose an adap-

tive approach based on establishing a merit function

to adapt the inter-class boundaries to the speciﬁc im-

age conditions. Figure 4-a shows a simpliﬁed exam-

ple of a 2D feature space including 4 binary proto-

types. The inter-class boundaries are symmetric with

respect to each prototype. In a real context, and image

of a pattern or any version translated or rotated will

have symmetry feature vectors SFV ( f

, f

) that never

reach certain areas close to the prototypes. These

areas can be founded computing the SFV for differ-

ent translated or transformed version of the same pat-

tern, and looking for f

min

and f

max

values. Figure 4-b

shows this forbiden areas. In this space, the distances

to the prototypes are distorted, so it would be desir-

able to adapt the boundaries between classes to this

situation.

To do this, a transformation of the the feature

space can be performed by normalising these fea-

tures using f

min

and f

max

values. In this new space

the null-class P1 disappears (every SFV has at least a

’1’ value), therefore this class should be treated sepa-

rately. In this new space, a boundary between classes

can be searched in a way that maximizes a merit func-

tion. To simplify the computations, we use orthogonal

boundaries deﬁned by a single parameter U , that rep-

resents the Uncertainty Boundary of Symmetry. We

studied several merit functions and, ﬁnally, propose

the distance ratio between the reported ﬁrst and sec-

ond classes after classifying the sample with respect

to binary prototypes using a NN classiﬁer. The result

is the boundary U

opt

that best separates the classes.

Moreover, instead of moving the inter-class bound-

aries, the problem is reformulated to modify the class

prototypes into new values (H, L) ∈ [0, 1] that are

symmetrical with respect to the middle value U (Fig-

ure 4-c). Finally, the closest class to new prototypes

(H, L) and the null-class P1 are disambiguated (Fig-

Figure 4: (a) A 2D feature space including binary proto-

types P1, P2, P3 and P4. (b) Forbiden areas around the

prototypes. (c) Normalized space to f

min

and f

max

. Adap-

tion of class boundaries. (d) Prototype adaption and ﬁnal

P1 disambiguation.

ure 4-d). The algorithm is as follows:

Step 1 - The symmetry values are normalized between

the maximum and minimum values of SFV :

SFV

= ( f

, f

, ..., f

);

−min(SFV )

max(SFV )−min(SFV )

; 1 ≤ i ≤ 12

(3)

In this way we assume that every image has one

symmetry and one non-symmetry value, discharging

the P1 class and resulting in a 16-class problem.

VISAPP 2011 - International Conference on Computer Vision Theory and Applications

Step 2 - The original prototypes are transformed into

(H, L) prototypes for each of 16 classes. These values

are deﬁned with respect to parameter U as in Eq. 4:

U ∈ ℜ, 0 ≤ U ≤ 1

U ≥ 0, 5 ⇒ H = 1, L = 2 ·U − 1

U < 0, 5 ⇒ L = 0, H = 2 ·U

(4)

Step 3 - For each U, ranging from 0.2 to 0.8, the H

and L limits are computed using Eq. 4 and a near-

est neighbour classiﬁcation is performed using SFV

and the resulting prototypes. The merit function is the

distance ratio between the reported ﬁrst and second

classes. After repeating steps 2-3 for all U values, the

value (U

opt

) that maximizes the merit function is se-

lected, and the corresponding class is also tentatively

selected (candidate class).

Step 4 - Finally, we disambiguate the candidate class

from the previously excluded P1 class. To achieve

this, we again re-classify the SFV but only using the

P1 and candidate classes. The nearest class is selected

as the ﬁnal result.

4 EXPERIMENTS

As indicated in (Liu et al., 2010), without a systematic

evaluation of different symmetry detection and classi-

ﬁcation algorithms against a common image set under

a uniform standard, our understanding of the power

and limitations of the proposed algorithms remains

partial.

As image datasets reported in literature were not

publicly available, we selected several wallpaper im-

ages from known websites, to carry out the compari-

son between the proposed ANNC and the reference

RBC methods. We picked out images from Wall-

paper dataset in (Joyce, 2007), Wikipedia dataset in

(Wikipedia, ), Quadibloc dataset in (Savard, ), and

SPSU dataset in (Edwards, 2009), resulting in a test

bed of 161 images. All images were hand-labelled to

make the ground truth. Also, an initial procces of lat-

tice extraction were performed to obtain the UL, using

wavelets (Agust

ı et al., 2008).

As the original RBC algorithm source code was

unavailable, we implemented it using the original

RBC decision tree reported in (Liu et al., 2004), but

using our SFV feature vectors, and binarising the fea-

tures using a ﬁxed threshold for all image datasets. A

threshold of 0.7 was obtained as the average of the

better classiﬁcation results when all the threshold val-

ues are explored.

The results obtained with RBC, NNC and ANNC

classiﬁers are shown in Table 3. For the shake of

brevity, we only put here the percentage of success-

ful classiﬁcation, i.e. accuracy or precision results.

Table 3: Experimental classiﬁcation results from RBC,

NNC, and ANNC.

Collection Sub-set RBC NNC ANNC

Wallpaper 100 94,12 100

Wikipedia 56,6 60,38 54,72

WikiGeom 88,24 76,47 94,12

Quadibloc 69,57 73,91 82,61

Quad0102 62,07 79,31 79,31

Quad03 88,24 58,82 88,24

SPSU 45,37 52,75 54,63

The ﬁrst image collection is Wallpaper, a standard

collection reported in previous works. In this case,

both RBC and ANNC methods obtain a 100% of suc-

cess. The RBC achieves the same result as reported in

(Liu et al., 2004), which means that our implementa-

tion of this algorithm has similar results as the original

implementation.

To take into account the varying complexity of the

images, we separate the next two collections in sub-

sets. In the WikiGeom dataset, which is a sub-set of

Wikipedia formed by strongly geometrical patterns,

the ANNC and NNC results outperformed the RBC.

Figure 5 shows some examples of this collection. In

the case of the entire Wikipedia collection, which in-

clude other distorted images, a decrease in results is

evident.

Similar results were obtained with Quadibloc im-

age collection, which is of intermediate complex-

ity. We studied it as two subsets: one formed by

sketches over uniform background (Quad0102), and

other (Quad03) is constituted by more complex mo-

tives with many highly contrasted colours are present.

The ANNC obtains near 80% of success rate with this

collections, clearly outperforming the NNC and RBC.

The worse results were obtained with the more

complex images in the SPSU collection. The Figure

6 show some examples of this collection. In this case,

all results are below 60%. This lower values are due

to the existence of noise and imprecise details (hand-

made) in the images. Also, they exhibit several rep-

etitions and illumination artifacts, which suggest the

necessity of pre-processing. It is remarkable that the

ANNC algorithm is still 10 points up that RBC algo-

rithm for this complex dataset.

5 CONCLUSIONS

This paper had presented a proposal for an adaptive

nearest neighbour classiﬁer (ANNC) able to classify

COMPUTATIONAL SYMMETRY VIA PROTOTYPE DISTANCES FOR SYMMETRY GROUPS CLASSIFICATION

Figure 5: Wallpaper pattern examples in the wikiGeom collection.

Figure 6: Wallpaper pattern examples in the SPSU collection.

repetitive 2D pattern images into symmetry groups.

The feature vector is composed of twelve symmetry

scores, computationally obtained from the image gray

level values. The procedure uses a Sum-of-Absolute-

Differences approach and normalize these values by

the translational symmetry, the only symmetry that

image certainly contains. A distance-based Nearest

Neigbour classiﬁer is then used to classify the image

into a symmetry group. A main issue is the use of bi-

nary class prototypes to represent the 17 PSG classes.

However, the absence of symmetry is never com-

puted as ’0’, nor the presence of symmetry is com-

puted as ’1’, even assuming perfect image conditions

(synthetic images). The dynamic range of the sym-

VISAPP 2011 - International Conference on Computer Vision Theory and Applications

metry values ( f

) is extremely variable, depending on

the speciﬁc conditions of each image. The binary

Rule Based Classiﬁer (RBC) and the Nearest Neig-

bour classiﬁers behave poorly in that situation. This

leads to the use of some adaptive approach. The

proposed ANNC classiﬁer is based on establishing a

merit function to adapt the inter-class boundaries to

the speciﬁc image conditions, and it is reformulated

as an adaptation of prototype feature vectors from bi-

nary values (0, 1) to adjusted (H, L) values. This clas-

siﬁer is non-parametric, so there is no need to adjust

the parameters involved, and it is also non-supervised,

so no learning stages are needed. The experimental

results show the limits of previous reported method

(RBC) when the image conditions are not established

and the need to adapt to these conditions. The ANNC

clearly outperforms the other methods, even with very

complex image collections.

As future work we are now on looking for a new

way of computing the symmetry features, because

the used approach seems to have a limited sensitiv-

ity. We are also extending the method to colour im-

ages. Moreover, the results can be useful in recovery

tasks using an extended version of ANNC. Finally, we

will concentrate on the matter of test beds: a standard

database does not exist in bibliography.

ACKNOWLEDGEMENTS

This work is supported in part by spanish project VIS-

TAC (DPI2007-66596-C02-01).

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