CLASSIFYING AND COMPARING REGULAR TEXTURES FOR

RETRIEVAL USING TEXEL GEOMETRY

Junwei Han and Stephen J. McKenna

School of Computing, University of Dundee, Dundee, DD1 4HN, U.K.

Keywords: Regular texture, Texel geometry, Texel classification, Texel comparison.

Abstract: Regular textures can be modelled as consisting of periodic patterns where a fundamental unit, or texel,

occurs repeatedly. This paper explores the use of a representation of texel geometry for classification and

comparison of regular texture images. Texels are automatically extracted from images and the distribution

of texel shape and orientation is modelled. The application of this model to image retrieval and browsing is

discussed using examples from a database of art and textile images.

1 INTRODUCTION

Regular textures can be modelled as consisting of

periodic patterns where a fundamental unit (texel)

occurs repeatedly. Texture periodicity analysis has

attracted much attention recently and has been used

for texture tracking (Lin et al., 2007), synthesis

(Charalampidis, 2006), and retrieval (Liu et al.,

1996; Lin et al., 1999; Lee et al., 2005).

In common with much of the previous work, this

paper focuses on the study of so-called wallpaper

patterns. There exist 17 wallpaper groups which

together account for all patterns generated by two

linearly independent vectors (Liu et al., 2004). Here,

regular textures generated by translation only are

considered, as shown in Figure 1. A pair of vectors

with shortest length (two linearly independent

directions), (

t ,

t ) define a parallelogram which is

called the texel. The texel repeatedly tiles the image

to form a lattice structure.

t and

t define the size,

shape, and orientation of the texel.

Texel extraction is key to understanding regular

texture. Starovoitov et al. (1998) used features

derived from co-occurrence matrices to extract the

texel. Charalampidis (2006) achieved this in the

frequency domain based on the assumption that

fundamental frequencies hold the basic structure

information of regular texture. Lin et al. (1997)

obtained texels by detecting salient peaks in the

autocorrelation (AC) function of a texture image.

Liu et al. (2004) extended the work of Lin et al.

(1997) by adopting more dominant peaks of the AC

function.

Figure 1: A wallpaper pattern example with its two

placement vectors and lattice structure.

Several applications are based on the results of

texel extraction from regular texture. Chetverikov

(2000) and Leu (2001) measured the regularity

degree of images using features derived from the AC

function and similarity among texels, respectively.

The regularity measurement can be applied to

classify regular and irregular texture images. Texture

image retrieval and browsing systems have been

proposed in which the features used are related to

texture periodicity (Liu et al., 1996; Lin et al., 1999;

Lee et al., 2005). Charalampidis (2006) implemented

texture synthesis using extracted texels. Lin et al.

(2006) designed a geometric regularity score that

depended on both the magnitudes and directions of

t and

t to evaluate various texture synthesis

algorithms. Recently, Hays et al. (2006) and Lin et

al. (2007) extended regular texture models to extract

and track texels of near-regular texture, respectively.

As can be seen from Figure 1, texel geometry

indicates the spatial arrangement of a regular

texture. Lin et al. (2006) adopted texel geometric

information for the purpose of comparing

synthesized texels and original texels. The

347

Han J. and J. McKenna S. (2009).

CLASSIFYING AND COMPARING REGULAR TEXTURES FOR RETRIEVAL USING TEXEL GEOMETRY.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 347-354

DOI: 10.5220/0001792703470354

 SciTePress

comparison was based on a Euclidean distance

without taking the intrinsic distribution of texels into

consideration.

This paper presents a method that takes

advantage of the geometric information from texels

to retrieve and browse images. Firstly, texels are

automatically extracted from regular texture images.

Then, aspects of the texel geometry are represented

as feature vector. Based on the distribution of a

collection of images in the resulting feature space,

clusters are defined such that each cluster

corresponds to a type of texel. Each cluster is

modelled as Gaussian and Bayes’ rule is used to

estimate the probability that a regular texture has a

certain texel type. The estimated distributions are

also used to measure similarity between texels.

Finally, the proposed techniques are applied to

perform image retrieval and browsing.

The main contributions of this paper are: 1) A

3D representation is proposed to characterize texel

geometry and embody the spatial arrangement

information of regular texture in a manner that is

invariant to translation and scaling in the image

plane. 2) Texel clusters are defined and modelled

based on the distribution of a collection of data. 3)

Instead of using Euclidean distance, texel

comparisons are made based on the probabilities that

the image belongs to each cluster and the intrinsic

cluster distributions. Finally, we show how these

methods can be applied to image retrieval and

browsing.

The rest of the paper is organized as follows.

Section 2 summarises the texel extraction algorithm.

Section 3 proposes a model of texel types based on

the distribution of a collection of images. Section 4

applies the model to image browsing and retrieval.

Experiments are presented in Section 5. Finally,

conclusions are drawn in Section 6.

2 TEXEL EXTRACTION

A previously published method (Han et al., 2008)

was used to extract texels, i.e. to estimate (

t ,

t ).

The algorithm is described here briefly for

completeness.

The texel extraction algorithm contains two

steps: texel hypotheses generation and hypothesis

comparison. The first step begins by computing the

AC function. Peaks in AC functions are always

associated with texture periodicity. Following the

ideas of Lin et al. (1997) and Liu et al. (2004),

salient AC peaks are selected and used to obtain

texels. Changing the number of peaks considered

can result in different texel candidates.

The second step compares all of the texel

candidates obtained from the first step using a

Bayesian model comparison framework. Let

an image and

),(

≡

H denote a texel hypothesis

for

H the

k in a set of hypotheses, and

a statistical model defined based on

H with

parameters

. Texel extraction can be formulated

as choosing the most probable texel hypothesis

given the image. By Bayes’ theorem, the posterior

probability is proportional to the likelihood of the

hypothesis times a prior:

(| )( )

(|) (|)()

()

kkk

pI H pH

pH I pI H pH

=∝ (1)

In the absence of prior knowledge favouring any

particular hypothesis, the prior is taken to be

uniform. For each

H , we define a unique

M deterministically so )|(

HMp is a delta

function. Hence,

(|) (|) (|,)(|)

kkkkkkk

pH I pI M pI M p M d

θθ

∝=

∫

(2)

The integral in Eq. (2) can be approximated

using Bayes Information Criterion (BIC). The details

of BIC approximation can be found in Raftery

(1995). The BIC for the model is:

( ) log ( | , ) ( / 2)log log ( | )BIC M p I M d N p I M

=− + ≈−

(3)

where

d is the number of parameters and

is a

maximum likelihood parameter estimate.

The hypothesis with the model that has the

largest marginal likelihood is selected. Using the

BIC approximation, hypothesis

H is selected by

)}({argmin)|(argmax

MBICIHpk ==

(4)

The texel model

M should be able to account

for both regularity from periodic arrangement and

statistical photometric and geometric variability.

Here a Gaussian with covariance matrix of the form

was used to model a texel’s appearance. The

reader is referred to Han et al. (2008) for further

details.

3 TEXEL GEOMETRY

This paper focuses on modelling the geometry of a

texel, (

t ,

t ), and not its pixel values. We opt for a

representation that is scale invariant since the

physical scale of the imaged objects is unknown.

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

348

The following three features are used to describe the

spatial arrangement:



: the angle between

t and the image x-axis;



: the angle between

t and

t ;



: the ratio of lengths, i.e.

=r .

Note that the angle between

t and the x-axis is

not larger than the angle between

t and the x-axis,

by construction. Figure 2 shows an example. The

value of

ranges from 0 to 90 degrees.

t is the

texel vector that subtends the smallest angle with

t ,

and that angle is

. The value of

for a wallpaper

pattern always lies between 60 and 90 degrees.

Figure 2: An example of the geometry of a texel.

The three-dimensional feature vector ),,( r

automatically extracted from each regular texture

image using the method described in Section 2. A

distribution of 200 images in this 3D feature space is

shown in Figure 3. See Section 5 for details of the

dataset. Inspection of this distribution suggests

clusters. Each cluster can be considered to

correspond to a type of texel. Specifically, five

clusters of texels might be defined according to the

following five rules:

 Cluster 1: texels are rectangles with

90 ,0 ≈≈

φα

;

 Cluster 2: texels are parallelograms with

90 ,45 <≤

φα

;

 Cluster 3: texels are parallelograms with

90 ,0 ≈>

φα

;

 Cluster 4: texels are parallelograms with

90 ,45 <>

φα

;

 Cluster 5: texels are parallelograms with

90 ,0 <≈

φα

Any image in the dataset can be classified into a

cluster based on the defined rules. However, a model

of the cluster

distributions is more useful, enabling

the clusters to be parameterised and meaningful

texel similarity measures to be defined. Each cluster

can be modelled as a three-dimensional Gaussian

distribution with a probability density function

3/2 1/2

(| ) (2) | |

exp ( ) ( )

ii i

−−

−

⎧

⎫

⋅−− −

⎨

⎬

⎩⎭

x Σ

x μΣx μ

(5)

where

),,( r

x denotes the feature vector of an

image,

}5,4,3,2,1{,

∈

, denotes the cluster index

or class,

μ denotes the mean for class i, and

denotes the covariance matrix for class

i. The

parameters

μ and

Σ can be estimated using

maximum likelihood. The class posterior probability

can then be estimated via Bayes’ theorem,

)(

)()|(

)|(

CPCP

= (6)

where the prior

)(

CP can be estimated from the

frequencies of the classes in the data.

4 IMAGE BROWSING AND

RETRIEVAL

Due to the rapidly growing number of digital images

in our lives, there is a great need for effective image

retrieval techniques. Content-based image retrieval

using image features such as color, shape, and

texture can be effective when the user has a query

image to hand. However, when the user’s intention

is ambiguous, image browsing can be more useful.

Browsing supposes that the images can be

categorized and ordered in meaningful ways. In the

case of retrieval and browsing of images exhibiting

regular texture, the spatial arrangement is obviously

quite an important feature. In this section, we

illustrate how the technique for describing and

modelling texel geometry (the spatial arrangement

of regular texture) can be applied to content-based

retrieval and browsing.

CLASSIFYING AND COMPARING REGULAR TEXTURES FOR RETRIEVAL USING TEXEL GEOMETRY

349

Figure 3: A distribution of 200 regular texture images in the 3D texel feature space.

4.1 Browsing a Texel Class

One approach to organising an image database for

browsing is to categorise the images and to then

display images within a category in a meaningful

way.There are then two problems: (i) how to

categorise images, and (ii) how to lay out images

within a category meaningfully for display. It is

proposed that regular texture images can be

categorised according to texel geometry. As shown

in Figure 3, data points within a cluster tend to be

scattered along a one-dimensional trend. This

corresponds to the direction of maximal intra-class

variance which is given by the principal component

of the class distribution. This direction gives a good

feature for intra-class discrimination and motivates

projecting data onto these principal components.

Ranking images according to the projected values

will reflect the intra-class variation of texel

geometry. More formally,

1. Given a set of training images from a texel

class, estimate the class mean

μ , the

covariance matrix

Σ , and the eigenvector

v of this matrix that corresponds to the

largest eigenvalue

2. For each test image from the same texel

class, project the texel

),,( r

=x onto the

first eigenvector:

μxv )( −=y .

3. List the images in ascending order of their

value.

Figure 4 shows an example of browsing an

image class. The original image is shown first,

followed by its lattice structure. In this category,

there were a total of 60 images. Due to limited

space, only a few images are shown. The first row

shows the three images placed at the front of the

ranking list, the second row shows the middle three,

and the third row shows the last three images. Texels

of this image class differ mainly in the value of

which varies from approximately 60 degrees to

approach 90 degrees. Projecting the image data to

the trend of the first principal component retains the

‘most important’ variation. Images in this category

are thus sorted in order of increasing

4.2 Image Retrieval

Key to image retrieval and browsing is to measure

similarity between images, whether between a query

and the database for retrieval, or between images in

the database for structuring the data for indexing and

visualisation. Consider the case of query-by-

example in which a query image

Q is to be

compared to an image

from the database.

Assume that

has been classified as belonging to

the class

}5,4,3,2,1{, ∈jC

. The similarity between

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350

and

is estimated as follows.

1. Calculate the probability

)|( QCP

that

belongs to class

. (See Eqs. (5) and (6)).

2. Project

Q and

onto the principal

component of class

C to obtain

μv )(

jjQ

Qy −= and

μv )(

jjA

Ay −= .

3. The similarity of

Q to

is computed as:

2/1

)2( where

)|(

)(

exp

QCP

λπ

⎟

⎠

⎞

⎜

⎝

⎛

−

−=

(7)

The above processing is repeatedly performed to

every image in the database to yield a similarity to

the query. Then, the images are ranked in decreasing

order of similarity to the query. The similarity

measure in Eq. (7) takes into account the probability

that the query belongs to the same class and the

distance between the images in that class

(appropriately scaled).

Figures 5 and 6 show examples of the proposed

image retrieval algorithm. In each of these Figures,

the query image is shown at the top-left, and the top

eight returned images are shown. The images are

ordered from left to right and from top to bottom.

Recall that the texel geometry is represented in a

way that is scale invariant. Therefore, the similarity

measure is in terms of shape and orientation. As can

be seen, the returned images have their basic texture

units repeated in similar ways to the query images.

5 EXPERIMENTS

Three experiments were performed to evaluate the

proposed methods. The first experiment tested the

performance of the texel extraction algorithm. The

second experiment tested the ability of the Gaussian

cluster models to yield correct classification of

texels. The final experiment explored the ability of

the principal components to represent the clusters.

A dataset of 200 regular texture images was used

for evaluation, comprising 147 images of textiles

from a commercial archive and 53 images taken

from three public domain databases (the Wikipedia

Wallpaper Groups page, a Corel database, and the

CMU near-regular texture database). The images

ranged in size from 300

× 225 pixels to 2648

1372

pixels. The number of texel repeats per image

ranged from five to a few hundreds. This data set

includes images that are challenging because of (i)

appearance variations among texels, (ii) small

geometric deformations, (iii) texels that are not

distinctive from the background and are large non-

homogeneous regions, (iv) occluding labels, and (v)

stains, wear and tear in some of the textile images.

5.1 Evaluation of Texel Extraction

Two volunteers (one male and one female)

qualitatively scored and rank ordered the algorithms.

In cases of disagreement, they were forced to reach

agreement through discussion. (Disagreement

happened in very few cases). The observers were

shown extracted texels overlaid on images and were

asked to label each texel as obviously correct (OC),

obviously incorrect (OI), or neutral. They were to

assign OC if the texel was exactly the same or very

close to what they expected, OI if the result was far

from their expectations, and neutral otherwise. In

our texel extraction algorithm, variance of the

Gaussian model was the only free parameter and it

was set as

100

. The numbers of OC, OI, and

neutral results were

164, 17 and 19, respectively.

Thus, the accuracy of texel extraction was

164/200=82%. Figure 7 shows some example

results.

5.2 Evaluation of Gaussian Model

A classification experiment was performed to assess

the suitability of the assumption of Gaussian

clusters. Images were classified as belonging to the

cluster with the largest posterior probability as

computed using Equations (5) and (6).

The data set of

200 images was divided into two

disjoint sets of

100 images each. One was used as a

training set and the other as a test set. The

experiment was then repeated after switching the

training and test sets. Training data and ground truth

were labelled using the rules in Section 3.2. The

classification rates for the two test sets were

91%

and

96% giving an average rate of 93.5%. The

confusion matrix averaged over the two test sets is

shown in Table 1. Regular textures from classes 2, 3,

and 4 were more likely to be misclassified, as would

be expected from inspection of Figure 3.

5.3 Evaluation of Texel Comparison

It was proposed that texels be represented by

projection onto their class-specific principal

component. The intra-class distribution is thus

modelled as a 1D Gaussian. An experiment was

CLASSIFYING AND COMPARING REGULAR TEXTURES FOR RETRIEVAL USING TEXEL GEOMETRY

351

Figure 4: Image browsing example for class 4.

Figure 5: Query-by-example based on texel geometry. The query is top-left followed by the seven best matches.

Figure 6: Query-by-example based on texel geometry. The query is top-left followed by the seven best matches.

performed to explore the effect of this projection.

Texels from class i can be generated from this model

by:

avμx

= (8)

where

a is an appropriately set weight. Weights

with large magnitudes result in texels far from the

mean. In practice, data will fall in a range such as

λλ

33 ≤≤− a (9)

where

is the eigenvalue for eigenvector

v .

Table 2 shows texels synthesised from each of the

five classes by setting

λλ

2,,0 ±±=a .

Table 1: Confusion matrix for texel classification.

Predicted

True

29.5 0.5 0.0 0.0 0.0

0.0 22.0 2.0 0.0 0.0

0.0 1.5 21.5 0.0 0.0

0.0 1.5 1.0 10.0 0.0

0.0 0.0 0.0 0.0 10.5

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352

Table 2: Synthetic texels generated by the model.

-2

Class 1

Class 2

Class 3

Class 4

Class 5

Figure 7: Some results from the texel extraction algorithm.

It can be seen that the major mode of variation for

class 1 was the ratio of the lengths of

t and

t .

The major mode of variation for class 3 combined

the ratio of the lengths of

t and

t , and the

direction of

t . The major mode of variation for

classes 2 and 4 involved all three features. These

synthetic data suggest that the proposed model is

able to capture the variability of each class

effectively.

6 CONCLUSIONS

In this paper, a systematic study of the texel

geometry of regular textures has been presented. A

fully automatic algorithm using Bayesian model

comparison was used to extract texels. A feature

vector defined on the obtained texel was proposed to

characterize the geometry of a texel. The distribution

of a set of regular texture images in the feature space

was modelled. The proposed model is easy to

implement and was applied to guide image browsing

and retrieval effectively. Experiments on a collection

of regular texture images have demonstrated the

promise of the approach.

Various extensions to this work would be

interesting to investigate in future work. 1) It would

be useful to analyse other regular texture data sets to

investigate the breadth of applicability of the

proposed clustering model. 2) Evaluations of image

retrieval and browsing should be conducted on a

large-scale database combining the proposed

technique with other features that model the

CLASSIFYING AND COMPARING REGULAR TEXTURES FOR RETRIEVAL USING TEXEL GEOMETRY

353

appearance of the texels. 3) The proposed work has

been applied to image retrieval and browsing in this

paper. We believe it can also be extended to help

fabric designers to categorize and manage their

digital archives, and provide them with interesting

sources to spark and fuel design inspiration.

ACKNOWLEDGEMENTS

This research was supported by the UK Technology

Strategy Board grant ``FABRIC: Fashion and

Apparel Browsing for Inspirational Content'' in

collaboration with Liberty Fabric, System

Simulation, Calico Jack Ltd., and the Victoria and

Albert Museum. The Technology Strategy Board is

a business-led executive non-departmental public

body, established by the government. Its mission is

to promote and support research into, and

development and exploitation of, technology and

innovation for the benefit of UK business, in order

to increase economic growth and improve the

quality of life. It is sponsored by the Department for

Innovation, Universities and Skills (DIUS). Please

visit www.innovateuk.org for further information.

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