![Fig. 7. Examples of suppression of contour perception by a grating: (a) the side of the triangle that is parallel to the bars of the grating does not pop out as the other two sides do and (b) part of the contour of the rectangle is “lost” in the grating [56].](/figures/fig-7-examples-of-suppression-of-contour-perception-by-a-2qvcj564.webp)
University of Groningen
Comparison of texture features based on Gabor filters
Grigorescu, Simona E.; Petkov, Nicolai; Kruizinga, Peter
Published in:
Ieee transactions on image processing
DOI:
10.1109/TIP.2002.804262
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2002
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2002). Comparison of texture features based on Gabor
filters.
Ieee transactions on image processing
,
11
(10), 1160-1167. https://doi.org/10.1109/TIP.2002.804262
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.
More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-
amendment.
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 09-08-2022
1160 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 11, NO. 10, OCTOBER 2002
Comparison of Texture Features
Based on Gabor Filters
Simona E. Grigorescu, Nicolai Petkov, and Peter Kruizinga
Abstract—Texture features that are based on the local power
spectrum obtained by a bank of Gabor filters are compared. The
features differ in the type of nonlinear post-processing which is ap-
plied to the local power spectrum. The following features are con-
sidered: Gabor energy, complex moments, and grating cell oper-
ator features.Thecapability of the corresponding operatorsto pro-
duce distinct feature vector clusters for different textures is com-
pared using two methods: the Fisher criterion and the classifica-
tion result comparison. Both methods give consistent results. The
grating cell operator gives the best discrimination and segmen-
tation results. The texture detection capabilities of the operators
and their robustnessto nontexture features are also compared.The
grating cell operator is the only one that selectively responds only
to texture and does not give false response to nontexture features
such as object contours.
Index Terms—Classification, complex moments, discrimination,
features, Fisher criterion, Gabor energy, Gabor filters, grating
cells, local power spectrum, segmentation, texture.
I. INTRODUCTION
V
ARIOUS features related to the local power spectrum of
images have been proposed in the literature and used in
one way or another for texture analysis, classification, and/or
segmentation. In most of these studies the relation to the local
spectrum is established through (intermediate) features that are
obtained by filtering the input image with a set of two-dimen-
sional (2-D) Gabor filters. Such a filter is linear and local. Its
convolution kernel is a product of a Gaussian and a cosine func-
tion. The filter is characterized by a preferred orientation and
a preferred spatial frequency. Roughly speaking, a 2-D Gabor
filter acts as a local band-pass filter with certain optimal joint
localization properties in the spatial domain and in the spatial
frequency domain [1]. Typically, an image is filtered with a set
of Gabor filters of different preferred orientations and spatial
frequencies that cover appropriately the spatial frequency do-
main, and the features obtained form a feature vector field that
is further used for analysis, classification, or segmentation.
Gabor feature vectors can be used directly as input to a clas-
sification or a segmentation operator or they can first be trans-
formed into new feature vectors that are then used as such an
input. In [2]–[8], for example, pairs of Gabor features that cor-
respond to the same preferred orientation and spatial frequency
Manuscript received December 19, 2000; revised May 31, 2002. The asso-
ciate editor coordinating the review of this manuscript and approving it for pub-
lication was Prof. Pierre Moulin.
S. E. Grigorescu and N. Petkov are with the Institute of Mathematics and
Computing Science, University of Groningen, Groningen, The Netherlands
(e-mail: simona@iwinet.rug.nl; petkov@cs.rug.nl).
P. Kruizinga is with Océ Technologies, Venlo, The Netherlands.
Digital Object Identifier 10.1109/TIP.2002.804262.
but differ in the value of a phase parameter are combined in a
quantity called the Gabor energy. In references [9], [10], com-
plex moments are derived from Gabor features. Finally, in ref-
erences [11]–[14] grating cell operator features, inspired by the
function of a special type of visual neuron, are computed using
Gabor features.
Since the type of “post-Gabor” processing in the above men-
tioned methods is different, it is interesting to evaluate the effect
of the various types of nonlinear post-processing on the useful-
ness of the resulting features regarding texture discrimination
and segmentation.
At this point, the question arises of how to measure the use-
fulness of different features. Several authors have made a com-
parison of the performance of various operators and features for
texture segmentation. Most of these studies are based on a clas-
sification result comparison. In this method, a segmentation al-
gorithm is applied to a feature vector field and the number of
misclassified pixels is used to evaluate the segmentation per-
formance and suitability of the features. While this method is
widely used for feature comparison [15]–[22], one should keep
in mind that it characterizes the joint performance of a feature
operator and a subsequent classifier.
In [12] and [13], we proposed a method that can be used to
compare the features only, regardless of any subsequent clas-
sification or segmentation operations. This method is based on
a statistical measure of the capability of a feature operator to
discriminate two textures by quantifying the separability of the
corresponding clusters of points in the feature space according
to the Fisher criterion. While this method of feature evaluation
excludes the influence of the classification phase and focuses on
the feature extraction operators only, one should be aware of the
fact that the choice of a specific evaluation method inevitably
introduces certain limitations. In particular, the use of Fisher
criterion implies that the mean and the variance of a feature
distribution are important and adequate—not necessarily com-
plete—characteristics of the involved distributions, an assump-
tion that has been customarily made in the literature [23]–[30].
None of the aforementioned evaluation methods can be gen-
erally considered as superior because each of them is informa-
tive in its own way and each has its limitations. Using them both
gives a more accurate picture of operators’ performance.
This study comes as a natural continuation of the work
presented in [13]. There an operator that models the texture
processing properties of the visual system of monkeys and,
most probably, of man—the grating cell operator—was intro-
duced and compared with other artificial operators that are
devised by man. The results showed that the features obtained
with the grating cell operator were the best ones for a test image
1057-7149/02$17.00 © 2002 IEEE
GRIGORESCU et al.: COMPARISON OF TEXTURE FEATURES BASED ON GABOR FILTERS 1161
material containing oriented textures. These results prompted
other questions: is this outcome due only to the orientation
selectivity properties of the Gabor filters or is also the type
of “post-Gabor” processing that matters; how do nonlinear
post-processing schemes devised with mathematical models of
the texture in mind perform in comparison with a scheme in-
spired by the mammal visual system? With this study we try to
answer these questions. We restrict the comparison to operators
based only on Gabor filters because we evaluated in a similar
way other types of operators elsewhere [31]. In this paper, we
do not address the issue of Gabor filters selection since this
subject has already been sufficiently treated in [32]. Finally,
we examine only those types of nonlinear post-processing that
were proposed in the literature. Hence, it is beyond the scope of
this paper to propose new types of nonlinearities or to combine
already proposed ones in order to obtain better features.
The paper is organized as follows: in Section II, we review the
linear Gabor filter and various operators based on it. The prop-
erties of the concerned operators with respect to texture discrim-
ination are compared in Section III using the Fisher criterion. In
Section IV, a number of texture segmentation experiments are
carried out and the properties of the considered operators are
assessed using the classification result comparison method. In
Section V, the robustness of the operators to nontexture input
stimuli is studied. The paper is concluded with a discussion in
Section VI.
II. T
EXTURE FEATURES BASED ON GABOR FILTERS
A. Gabor Filters
A number of authors used a bank of Gabor filters to extract
local image features [2], [4]–[6], [33]. Typically, an input image
, ( —the set of image points), is convolved
with a 2-D Gabor function
, , to obtain a
Gabor feature image
as follows:
(1)
We use the following family of Gabor functions
1
(for further
details we refer to [14] and [34])
(2)
where
and
In our experiments, we use two filter banks, one with symmetric
(
) and the other with antisymmetric [ ]
Gabor kernels. Each bank comprises 24 Gabor filters that are
the result of using three differentpreferred spatial frequencies of
23, 31, and 47 cycles per image and eight different equidistant
preferred orientations [
, ). This
1
Two-dimensional Gabor functions and their power spectra can interactively
be generated and visualized at http://www.cs.rug.nl/~imaging/ where a descrip-
tion of the Gabor filter and its relation to a specific type of neuron in the primary
visual cortex are available as well.
type of sampling of the spatial frequency domain takes into ac-
count the bandwidth properties of the Gabor filters used [13].
The application of such a filter bank to an input image results in
a 24-dimensional feature vector for each point of that image.
B. Gabor Energy Features
The outputs of a symmetric and an antisymmetric kernel filter
in each image point can be combined in a single quantity that
is called the Gabor energy. This feature is related to the model
of a specific type of orientation selective neuron in the primary
visual cortex called the complex cell [35] and is defined in the
following way:
(3)
where
and are the responses
of the linear symmetric and antisymmetric Gabor filters, respec-
tively. The result is a new, nonlinear filter bank of 24 channels.
The Gabor energy is closely related to the local power spec-
trum. The local power spectrum associated with a pixel in an
image is defined as the squared modulus of the Fourier trans-
form of the product of the image function and a window func-
tion that restricts the Fourier analysis to a neighborhood of the
pixel of interest. Using a Gaussian windowing function as the
one used in (2) and taking into account (1) and (3) the following
relation between the local power spectrum
and the Gabor
energy features can be proven:
(4)
C. Complex Moments Features
In [9] and [36], the real and imaginary parts of the complex
moments of the local power spectrum were proposed as features
that give information about the presence or absence of dominant
texture orientations.
The complex moments of the local power spectrum are de-
fined as follows:
(5)
where
The sum , called the order of the complex moment, is
related to the number of dominant orientations in the texture. In
[36], it is proventhat a complex moment of evenorder
has
the ability to discriminate textures with
dominant ori-
entations. More precisely, the moduli of the complex moments
give information about the presence or absence of dominant ori-
entations while their arguments specify which orientations are
dominant. In [36], the authors discuss the advantages of using
the real and imaginary parts of the complex moments as features
instead of their moduli and arguments.
In our experiments, we use as features the nonzero real and
imaginary parts of the complex moments of the local power
1162 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 11, NO. 10, OCTOBER 2002
spectrum. For each point in the image we compute the com-
plex moments of up to order 8 resulting in a set of 45 complex
values. From this set we select only the nonzero real and imag-
inary parts. It can be proven that the complex moments of odd
order are zero and that all complex moments
for which
are real. Moreover, and are real due to the
discretization of the frequency domain used in the computation
of the local power spectrum. This amounts to 43 nonzero real
values out of which only 24 are linearly independent because
. We use this set of 24 linearly independent values
computed for each point in the image as a feature vector asso-
ciated with that point. In fact, we apply a nonsingular linear
transform to the local power spectrum.
We compute the complex moments of up to order 8 in order
to obtain the same dimensionality—24—of the feature space as
in the experiments with the other types of feature. Taking only
moments of up to order 4 or 6 can be regarded as an implicit
feature space dimensionality reduction step. Such a step can im-
prove the separability of the feature clusters, but then this step
should also be applied to the other features. Since in the scien-
tific community there is no general agreement whether a space
dimensionality reduction step is a part of the feature extraction
phase or not, we chose to keep the dimensionality of the feature
space the same for all considered operators (see further Sec-
tion VI).
The local power spectrum features are obtained using the
same filter bank as in the computations of the Gabor energy
features and consequently have the same coverage of the spa-
tial frequency domain.
D. Grating Cell Operator Features
A different type of nonlinearity is applied in an operator that
is based on a computational model of a specific type of neuron
found in areas V1 and V2 of the visual cortex of macaque mon-
keys and called the grating cell [37], [38]. Grating cells are se-
lective for orientation but differ from the majority of orientation
selective cells found in the mentioned cortical areas in that they
do not react to single lines or edges, as for instance simple cells
(modeled by Gabor filters) or complex cells (modeled by Gabor
energy operators) do. A grating cell only responds when a set
of at least three bars of appropriate orientation and spacing is
present in its receptive field. The response increases with the
number of bars that cross the receptive field of the cell and satu-
rates at about ten bars. The grating cell operator was conceived
to reproduce the properties of grating cells as known from elec-
trophysiological researches [11]–[14]. Essentially, this operator
signals the presence of one-dimensional (1-D) periodicity of
certain preferred spatial frequency and orientation in 2-D im-
ages.
The grating cell operator, as proposed in [14], consists of two
stages. The first stage is constructed to respond at any position
to a set of three parallel bars of a given orientation and spacing
at that position. The second stage integrates the output of the
first stage in a certain surrounding to ensure that the output of
this second stage increases if more than three parallel bars are
present in the concerned surrounding. For further details on this
operator we refer to [13] and [14].
In our experiments we use a set of grating cell operators with
the same eight preferred orientations and three preferred spa-
tial frequencies as in the experiments with the other operators,
yielding a vector of 24 features in each point of the image.
III. S
EPARABILITY OF CLUSTERS OF FEATURE VECTORS
In this section the feature extraction operators presented
above are compared from the point of view of their ability to
discriminate between different textures by means of the Fisher
criterion.
A. Comparison Method
The feature vectors computed in different points of a texture
image are not identical; they rather form a cluster in the multi-
dimensional feature space. The larger the distance between two
clusters that correspond to two different types of texture, the
better the discrimination properties of the texture operator that
produced the feature vectors.
In order to determine the distance between two clusters of
feature vectors, it is sufficient to look at their projections onto a
1-D space, i.e., a line, under the assumption that this projection
maximizes the separability of the clusters in the 1-D space. A
linear transform that, under certain conditions, realizes such a
projection was first introduced by Fisher [39] and is called the
Fisher linear discriminant function. It has the following form:
(6)
where
and are the means of the two clusters, is the
inverse of the pooled covariance matrix of the two clusters,
is
a feature vector, and
is its 1-D projection.
This projection of the feature vectors into the 1-D space maxi-
mizes the Fisher criterion [40], which measures the separability
of the two concerned clusters in the reduced space
(7)
where
and are the standard deviations of the distributions
of the projected feature vectors of the two clusters and
and
are the projections of the means and , respectively. The
Fisher criterion expresses in one single quantity the distance
between two clusters relative to their size. The larger the value
of the Fisher criterion computed for two clusters, the better the
separability of the two clusters.
Strictly speaking, the transform given by (6) need not nec-
essarily maximize the value of
according to (7) for arbitrary
distributions. It, however, does so for a Gaussian distribution of
texture features, an assumption that has frequently been made
and claimed to hold in literature (see e.g., [23]–[30]). In the case
of non-Gaussian distributions, one can think of the transform
given by (6) as a first order approximation of the transform that
maximizes the quantity
in (7).
Widely used in statistics, Fisher criterion has also been
used for various purposes in the field of image processing and
computer vision: filter design [41], texture classification [21],
[42]–[45], and feature space dimensionality reduction [46].
Only recently, this criterion has been applied to the evaluation
of texture feature extraction operators [13].
GRIGORESCU et al.: COMPARISON OF TEXTURE FEATURES BASED ON GABOR FILTERS 1163
Fig. 1. Nine test images of oriented textures.
B. Results
We evaluated the performance of the operators presented in
Section II according to the Fisher criterion by looking at the
pair-wise separability of the feature clusters corresponding to
nine test textures (Fig. 1).
Whilethenumberoftestimagesusedislimited,onehastopoint
out that the only aspect that was taken into account in selecting
them is that the textures show a certain degree of “orientedness”
which is to guarantee that (some of) the Gabor filters employed
will respond. Further, no special attention was paid to selecting
thesetestimagesandtherearenoreasonstothinkthatthechoiceis
infavorofanyofthefeatureextractionmethodspresentedabove.
The pair-wise separability of the feature clusters corre-
sponding to the nine test textures was measured as follows.
The pooled covariance matrix was calculated for each pair of
images using 1000 sample feature vectors from each image.
Then the feature vectors were projected on a line using the
corresponding Fisher linear discriminant function and the
Fisher criterion was evaluated in the projection space. For
brevity, only essential statistics of the 36 Fisher criterion values
computed for each operator are given here (see Fig. 2).
The values obtained for the Gabor energy features are good.
The mean value of 6.33 says that there is practically no overlap
between two clusters. The worst case scenario, described by
the minimum value of 2.35 corresponds to a cluster overlap of
less than 2.5% (assuming Gaussiandistribution).The results ob-
tained with the Gabor energy features are remarkable, if one
compares them with the linear Gabor features or with the thresh-
olded Gabor features [47], [48]. Our experiments showed that
Gabor energy features, involving only a simple type of post-pro-
cessing, perform better than the linear and thresholded Gabor
features by an order of magnitude.
The separability achieved for the complex moments features
is smaller than the one achieved with the Gabor energy features.
Thisresult isdue tothefactthatthe complexmomentswere com-
puted from the local power spectrum and not from the Gabor en-
ergy features. The nonlinear dependence between the Gabor en-
ergyandthe localpowerspectrum(4)leads evidentlytodifferent
degrees of separability of feature vector clusters. As already re-
Fig. 2. Boxplot representation of the distribution of the Fisher criterion values
obtained with different texture operators.
ported in [31],using the square root in the post-processing phase
following the filtering improves the separability of the feature
clusters in terms of Fisher criterion. As a possible explanation of
this result, let us consider a particular case of two 1-D stochastic
variables
and . For simplicity, we assume that and
that
can take only two values and ( ) with equal
probability. The value of Fisher criterion in this case is
. Now, if we consider the squares and of
the two stochastic variables,the value of the Fisher criterion will
be
andit caneasily be shownthat
.In other words, thevalueofthe Fisher criterion
computed for thetwo stochastic variablesis larger than the value
of the Fisher criterion computed for their squares. A similar situ-
ation obviously occurs with the features derived from the Gabor
energyand its square, the local power spectrum. In this way, fea-
turesderiveddirectlyfromthelocalpowerspectrumformclusters
that are less separable than the clusters obtained with Gabor en-
ergy feature vectors.
Computing the complex moments of the local power spec-
trum can itself not improve the separability of the feature vector
clusters obtained from the local power spectrum. As already
mentioned in Section II-C above, the computation of the com-
plex moments of the local power spectrum is a nonsingular
linear transform of the local power spectrum. Taking in con-
sideration (6) and (7) it can be proven that the value of Fisher
criterion is not affected by such a transform.
For any pair of texture images, the inter-cluster distance com-
puted using the grating cell operator features is considerably
greater than the inter-cluster distance computed with any of the
other operators. The minimum value of the Fisher criterion ob-
tained for this type of feature is 5.44. If (in a first approximation)
we assume a Gaussian distribution for the feature clusters, the
theoretical cluster overlap corresponding to a Fisher criterion
value of 5.44 will be less than 0.01%, corresponding to a mis-
classification chance of one on ten thousand pixels.
IV. A
UTOMATIC TEXTURE SEGMENTATION
In this section, the feature extraction operators presented in
Section II are compared in the classical way, i.e., on the basis