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1. Texture
1.1 Introduction
Texture is a measure of the variation of the intensity of a surface, quantifying
properties such as smoothness, coarseness and regularity. It's often used as a region
descriptor in image analysis and computer vision.
The three principal approaches used to describe texture are statistical, structural and
spectral. Statistical techniques characterise texture by the statistical properties of the
grey levels of the points comprising a surface. Typically, these properties are
computed from the grey level histogram or grey level co occurrence matrix of the
surface.
Structural techniques characterise texture as being composed of simple primitives
called "texels" (texture elements), that are regularly arranged on a surface according
to some rules. These rules are formally defined by grammars of various types.
Spectral techniques are based on properties of the Fourier spectrum and describe
global periodicity of the grey levels of a surface by identifying high energy peaks in
the spectrum.
Texture is a repeating pattern of local variations in image intensity. Texture cannot be
defined for a point
Figure 1:Texture Definition
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For example, an image has a 50% black and 50% white distribution of pixels.
Figure 2: Texture Definition with three images
Three different images with the same intensity, distribution, but with different
textures.
Texture consists of texture primitives or texture elements, sometimes called texels.
– Texture can be described as fine, coarse, grained, smooth, etc.
– Such features are found in the tone and structure of a texture.
– Tone is based on pixel intensity properties in the texel, whilst structure
represents the spatial relationship between texels.
If texels are small and tonal differences between texels are large a fine texture results.
If texels are large and consist of several pixels, a coarse texture results.
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2. Texture Analysis
2.1 Introduction
Texture analysis and synthesis is very important for computer graphics, vision, and
image processing. During analysis, an example texture is summarized into a nonlinear
deterministic function using tree structure vector quantization. An output texture,
initially random noise, is then synthesized from this estimated function.
Many applications in computer graphics, vision, and image processing can benefit
from a texture analysis and synthesis algorithm. For example, textures have long been
used to decorate object surfaces in computer rendered images. However, natural
textures are often difficult to generate manually; therefore an algorithm to synthesize
a large texture from a small scanned patch will be desirable. In machine vision, the
ability to discriminate, segment, or classify textures is essential for tasks such as
automatic inspection. Natural texture images are also notoriously hard to compress
due to their high frequency contents, and compact representation may be achieved via
a good texture analysis algorithm. Despite these wide applicability, however, an
algorithm that is both efficient and capable of representing all the texture properties
has yet to come.
2.2 Texture Model
Texture is a common visual experience. However, a precise definition of texture is
difficult to formulate. Traditionally, textures have been categorized as either
structural or stochastic. A structural texture is characterized by a set of primitives
(texons) and placement rules. For example, a brick wall texture is generated by tiling
up bricks (primitives) in a layered fashion (the placement rule). A stochastic texture,
on the other hand, do not contain explicit primitives (such as grass and sand). Since
most natural textures are neither complete stochastic nor structural, we attempt to
characterize both kinds of textures with a single model. The probability distribution
of the intensity value for a single pixel is completely determined by the intensity
values of pixels in its spatial neighborhood. More formally, let N(x, y) be the
collection of pixel values in the neighborhood of a pixel located at (x, y). Then the
value of pixel (x, y), I(x, y), can be represented as follows:
where F is a probability distribution that determines the pixel value of I(x, y) from its
neighborhoods N(x, y). The size of the neighborhood determines the “randomness” of
the texture. Textures with regular patterns such as tiled floor require large
neighborhoods, while small neighborhoods are sufficient for stochastic textures as
such sand beach. Given such a model, the goal of a texture analysis and synthesis
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algorithm is to estimate the parameters of F, and generate new textures by sampling
from it.
Figure 3 : The process/Algorithm in short
Texture analysis is an important problem since it conditions the quality of
segmentation and interpretation in lots of applications such as in the textile industry
or for satellite imaging. There exits two principal techniques in texture analysis.
The first approach consists in characterizing a texture by using some parameters such
as the well known attributes from the co occurrence matrix or the run-length matrix
,etc. Most of attributes were proposed to characterize textured regions of an image.
The main problem concerns the difficulty to know which attributes might be used
for an application because the efficiency of each type of attribute is often badly
known.
The second approach tries to define a texture model as a realization of a 2D random
field. Lots of texture models have been proposed in the literature but are usually used
for stochastic textures.
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3. Grey Level Co-Occurrence Matrix
3.1 Introduction
The role of automatic visual inspection has grown up during the last decade in the
same way as the computational power has increased and the prize of computation has
decayed. Visual inspection, especially, in the sense of visual quality control has
gained in popularity. Nowadays, several commercial applications are available e.g. in
paper mills in saw mills, and in rock industry. Most of these applications are based on
texture classification.
Visual texture contains variations of intensities, which form certain repeated patterns.
Those patterns can be caused by physical surface properties, such as roughness, or
they could result from reflectance differences, such as the color on a surface.
Differences observed by visual inspection are difficult to define in quantitative
manner, which leads to demand of defining texture using some features.
Among the current approaches used in image processing to describe texture, the so
called statistical approach is the widely used because it produces good results with
low computational costs. This method considers the distribution of gray levels and
their interrelationship. The pixel values are used to construct numerical structures
which are associated to the texture pattern of the image. This pattern is based mainly
on the inter-relationship between one pixel and its neighbors.
Grey Level Co-Occurrence Matrix (GLCM) is one of the best known texture Analysis
methods. GLCM estimates image properties related to second order statistics. Each
entry (i,j) in GLCM corresponds to the number of occurrences of the pair of grey
levels I and j which are a distance d apart in the original image.
Grey Level Co-Occurrence Matrices are of two types:

First order GLCM
Figure 4: 1st order GLCM
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
Second order GLCM
Figure 5: 2nd Order GLCM
The pixel pairs (i,j) are defined by distance and angle:
Figure 6: Distance, Angle Relationship
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3.2 Mathematical Framework of GLCM
A co-occurrence matrix is a square matrix with elements corresponding to the relative
frequency of
Occurrence of pairs of gray level of pixels separated by a certain distance in a given
direction. Formally, the elements of a GxG gray level co-occurrence matrix Pd for a
displacement vector d = (dx,dy) is defined as :
Pd (i,j ) = |{((r,s) , (t,v)) :I (r, s)=i, I (t, v)= j}|
where I (×,× ) denote an image of size NxN with G gray values, (r, s), ( t, v)Î NxN,
(t,v)=(r + dx, s + dy) and |. | is the cardinality of a set.
Haralick, Shanmugan and Dinstein proposed 14 measures of textural features which
are derived from the co-occurrence matrices, and each represents certain image
properties as coarseness, contrast, homogeneity and texture complexity. Those that
most commonly used are given below:
3.2.1 Entropy :
Entropy measures the disorder of an image and it achieves its largest value when all
elements in P matrix are equal [3]. When the image is not texturally uniform many
GLCM elements have very small values, which imply that entropy is very large.
Therefore, entropy is inversely proportional to GLCM energy.
3.2.2 Contrast:
Contrast feature is a measure of the image contrast or the amount of local variations
present in an image.
3.2.3 Angular Second Moment:
Energy, also called Angular Second Moment and Uniformity is a measure of textural
Uniformity of an image. Energy reaches its highest value when grey level distribution
has either a constant or a periodic form. A homogenous image contains very few
dominant grey tone transitions and therefore the matrix for this image will have fewer
entries of larger magnitude resulting in large value for energy feature. In contrast if
the matrix contains a large number of small entries, the energy feature will have a
smaller value.
3.2.4 Inverse Difference Moment:
Inverse difference moment measures image homogeneity. This parameter achieves its
largest value when most of the occurrences in GLCM are concentrated near the main
diagonal. IDM is inversely proportional to GLCM contrast.
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In all the above equations p(i,j) refers to the normalized entry of the co-occurrence
matrices. That is p(i,j) = Pd (i,j )/R where R is the total number of pixel pairs (i,j). For
a displacement vector d = (dx,dy) and image of size NxM R is given by (N-dx)(Mdy).
3.3 How to use GLCM?
Generally, the problem of texture discrimination based on statistical approach
consists on the analysis of a set of co-occurrence matrices. In this matrix, the indexes
of rows and columns represent the given range of the image gray levels, and the value
P(i,j) stored at the position (i,j) is the frequency that gray levels i and j occur with, at
a given distance and at a given direction. For instance, suppose we have the image
represented by its gray level matrix shown below Regarding the 0 degrees direction
and the distance 1, we will have the co occurrence matrix shown on the right. For
instance, the value of the co-occurrence P(0,1) = 8 was calculated by scanning the
gray level matrix and, for each pixel with value 0, its left and right nearest neighbors
were checked and P(0,1) was incremented whenever a value 1 was found.
Matrix 1: How to use GLCM?
Figure 7: 3D view of the Image from the Matrix1
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The figure above is a 3D image of the Grey Level Co-Occurrence Matrix of the
example shown above with d=1 and angle equal to 0 degrees (α = 0º )
After these calculation is done, the matrix is reduced by dividing each value by
normalization factors rendering a matrix whose sum is equal to 1. One co-occurrence
matrix is created for each pair direction-distance considered by the texture analysis.
Descriptor. Having a co-occurrence matrix, different properties of the pixel
distribution can be obtained by applying mathematical operations on the matrix
values. These operations are called descriptors. Each descriptor is related to a
particular visual feature about texture. The most common descriptors are already
described above ( Entropy, contract etc..)
Image Signature. This is a numeric value or a set of them, which can be used to
represent an image regarding one or more characteristics of it. For example, one
signature that can be assigned to an image is a vector of values obtained from the
application of the Energy descriptor over its set of co-occurrence matrices. The image
signature is generally used as parameter of comparison between two images.
3.4 Method in 4 steps:
Step 1: Obtain the co-occurrence matrices of each image. The gray levels of the
image are reduced to 16. For 0 degrees, 45 degrees, 90 degrees and 135 degrees
directions, and distances 1, 2, 3, 4 and 5, calculate the corresponding co-occurrence
matrix. This produces 20 matrices of 16x16 integer elements per image.
Step 2: Obtain the values for the chosen descriptors. For each co-occurrence matrix,
the value for the descriptor (or descriptors) is calculated. For each image, the resulting
descriptor values are stored in a matrix where the rows represent the directions 0
degrees, 45 degrees, 90 degrees and 135 degrees and the columns represent distances
1, 2, 3, 4 and 5.
Step 3: Generate image signatures. The image signatures are calculated from the
descriptor matrix by averaging the values obtained with the different distances for
each direction. Aiming at eliminating dependencies of image rotation, distances are
calculated with different "rotations" on the elements of the signatures and the smaller
one is adopted.
Step 4: Compare the images through their signatures. The signatures of two images
are compared and the distance between them is calculated, using the Euclidean
distance function on the elements of the signature.
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3.5 Grey Level Co-Occurrence Matrix Algorithm:



Count all pairs in which the first pixel has a value I, and its matching pair
displaced from the first pixel by d has a value of j.
This count is entered in the ith row and jth column of the matrix Pd[i,j]
Note that Pd[i,j] is not symmetric, since the number of pairs of pixels having
gray levels [i,j] does not necessarily equal the number of pixel pairs having
gray levels [j,i].

3.6Example of Image Retrival using Co-Occurrence Matrix
In this example Texture Analysis of a number of rock samples is done, using CoOccurrence Matrix
The original testing material consists of 168 rock texture images. The images in the
database are divided into 7 different categories, each containing 24 distinct images.
One image from each class can be seen in the figure below:
Figure 8: Rock Images for the Experiment
3.6.1 Normalization of the Test Set
Before computing the co-occurrence matrices the images in the database are
normalized. The idea is to overcome the effects of monotonic transformations of the
true image gray levels caused by variations of lightning, lens film and digitizers.
Normalization is done for all the images in the database by setting mean and standard
deviation to common values.
3.6.2 Splitting of the Images
Apparently some of the images do not contain only one texture (especially class 4).
Therefore, each of the original images is divided into 9 sub images resulting in a
database, which consists of 1512 images (216 images from each class). Retrieval
results were produced for both original and split database.
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3.6.3 Retrieval Procedure
This CBIR (content based image retrieval) system consists of two major parts. The
first one is feature
extraction, where a set of features is generated to represent the content of each image
in the database. The second task is similarity measurement, where a distance between
the query image and each image in the database is computed using their feature
vectors so that the N most similar images can be retrieved. The block diagram of the
system is presented in the figure below.
Figure 9: Retrival Procedure
Co-occurrence matrices are calculated for all the images in the normalized database.
GLCM is build by incrementing locations, where certain gray levels i and j occur
distance d apart from each other. To normalize GLCM, its values are divided by total
number of increments.
Features energy, entropy, contrast and inverse difference moment are calculated from
each co occurrence matrix and their values are saved in the feature vector of the
corresponding image. In order to avoid features with larger values having more
influence in the cost function (Euclidean distance) than features with smaller values,
feature values inside each feature class are normalized in the range [0,1].
The similarity between images is estimated by summing up Euclidean distances
between corresponding features in their feature vectors. Images having feature
vectors closest to feature vector of the query image are returned as best matches.
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The distance parameter can be optimized for each texture type based on the size of
textural elements. However, as the testing database consists of images having
different sized textural elements, and therefore, a common distance parameter is
rather difficult to find. Thus the retrival results are produced using the distance vector
d=[1,1], since it takes better into consideration also textures containing small textural
elements.
This experiment was carried on as follows: each image is extracted from the database
and considered as a query image. For each query image 20 best matches were
retrieved from the database. Ideally the goal is to have all the retrieved query images
belonging to the same class as the query.
Overall percentage of best matches coming from certain class is shown in Table 1 and
Table 2. Table 1 shows the results for normalized whole rock images, while Table 2
represents the results for normalized split rock images.
Table 1: Results for Normalized whole rock images
Table 2: Results for normalized split rock
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3.6.4 Evaluation of the results
As can be seen from Table 2 the retrieval results especially for class 1, but also for
classes 3, 5, 6 and 7 are quite reasonable. However, problems occur when considering
classes 2, and particularly class 4. Main reason for this might be that some images
inside these classes contain large areas of some other texture. To reduce this problem
original images are divided into 9 blocks and the retrieval procedure is applied for
them. Since these blocks usually consist of one texture, visual appearance of the
retrieval results is better. The results of the experiment are shown below:
Figure 10: Retrival Results
The Figure below shows the FFT values of the query images:
Figure 11: FFT values of query images
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3.6.5 Conclusions
It is known that co-occurrence matrix approach is an effective method in classifying
homogeneous stochastic textures. However, the results of this study indicate that cooccurrence matrix approach is also an effective method in similarity evaluation of
textures.
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4. Normalized GLCM
The elements of Pd[i,j] can be normalized by dividing each entry by the total
number of pixel pairs.
Normalized GLCM; N[i,j], defined by:
which normalizes the co-occurrence values to lie between 0 and 1, and allows
them to be thought of as probabilities.
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5. Problems with GLCM
One problem with deriving texture measures from co-occurrence matrices is how to
choose the displacement vector d.
–
The choice of the displacement vector is an important parameter in the
definition of the GLCM.
–
Occasionally the GLCM is computed from several values of d and the one
which maximizes a statistical measure computed from P[i,j] is used.
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