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these patches. First, texture specific features are ex-
tracted from each patch. A patch can then be repre-
sented by a feature vector. Similar patches will be
represented through similar features. Thus, images
patches are implicitly quantized into textons. Textons
are compared using the Bhattacharyya coefficient be-
tween their feature vectors. Section 3.1 describes the
features extracted from image patches. The algorithm
is presented in Section 3.2.
3.1 Texture Features
Before computing LTD between texture images, a set
of several image features is extracted from each patch
to obtain the texton representation. There are 9 fea-
tures extracted from patches, that are described next.
An interesting remark is that the more features are
added to the texton representation, the better the ac-
curacy of the LTD method gets. However, a lighter
representation, such as the one based on 9 features,
results in a faster and more efficient algorithm. One
may choose to add or remove features in order to ob-
tain the desired trade-off between accuracy and speed.
The texton representation based on the 9 features that
are about to be presented next gives state of the art ac-
curacy levels in several experiments presented in Sec-
tion 4.
The first two statistical features extracted are the
mean and the standard deviation. These two basic
features can be computed indirectly, in terms of the
image histogram. The shape of an image histogram
provides many clues to characterize the image, but the
features obtained from an image histogram are not al-
ways adequate to discriminate textures, since they are
unable to indicate local intensity differences.
One of the most powerful statistical methods for
textured image analysis is based on features extracted
from the Gray-Level Co-Occurrence Matrix (GLCM),
proposed in (Haralick et al., 1973). The GLCM is
a second order statistical measure of image variation
and it gives the joint probability of occurrence of gray
levels of two pixels, separated spatially by a fixed vec-
tor distance. Smooth texture gives co-occurrence ma-
trix with high values along diagonals for small dis-
tances. The range of gray level values within a given
image determines the dimensions of a co-occurrence
matrix. Thus, 4 bits gray level images give 16 × 16
co-occurrence matrices. Relevant statistical features
for texture classification can be computed from a
GLCM. The features proposed by (Haralick et al.,
1973), which show a good discriminatory power, are
the contrast, the energy, the entropy, the homogene-
ity, the variance and the correlation. Among these
features that show a good discriminatory power, LTD
uses only four of them, namely the contrast, the en-
ergy, the homogeneity, and the correlation.
Another feature that is relevant for texture analysis
is the fractal dimension. It provides a statistical index
of complexity comparing how detail in a fractal pat-
tern changes with the scale at which it is measured.
The fractal dimension is usually approximated. The
most popular method of approximation is box count-
ing (Falconer, 2003). The idea behind the box count-
ing dimension is to consider grids at different scale
factors over the fractal image, and count how many
boxes are filled over each grid. The box counting
dimension is computed by estimating how this num-
ber changes as the grid gets finer by applying a box
counting algorithm. An efficient box counting algo-
rithm for estimating the fractal dimension was pro-
posed in (Popescu et al., 2013). The idea of the algo-
rithm is to skip the computation for coarse grids, and
count how many boxes are filled only for finer grids.
LTD includes this efficient variant of box counting in
the texton representation.
The work of (Daugman, 1985) found that cells in
the visual cortex of mammalian brains can be mod-
eled by Gabor functions. Thus, image analysis by
the Gabor functions is similar to perception in the hu-
man visual system. A set of Gabor filters with differ-
ent frequencies and orientations may be helpful for
extracting useful features from an image. The lo-
cal isotropic phase symmetry measure (LIPSyM) pre-
sented in (Kuse et al., 2011) takes the discrete time
Fourier transform of the input image, and filters this
frequency information through a bank of Gabor fil-
ters. The work of (Kuse et al., 2011) also notes that lo-
cal responses of each Gabor filter can be represented
in terms of energy and amplitude. Thus, Gabor fea-
tures, such as the mean-squared energy and the mean
amplitude, can be computed through the phase sym-
metry measure for a bank of Gabor filters with various
scales and rotations. These features are relevant be-
cause Gabor filters have been found to be particularly
appropriate for texture representation and discrimina-
tion.
Finally, textons are represented by the mean and
the standard deviation of the patch, the contrast, the
energy, the homogeneity, and the correlation extracted
from the GLCM, the (efficient) box counting dimen-
sion, and the mean-squared energy and the mean am-
plitude extracted by using Gabor filters. These texton
features can be extracted from all images before com-
paring them with LTD. Thus, the LTD computation
can be divided in two main steps, one for texton fea-
ture extraction, and one for dissimilarity computation.
After the feature extraction step, features should be
normalized. In practice, the described features work
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