The paper proposes a method for constructing an informative neighborhood for modeling texture images. To describe the characteristic features of textures used assumptions underlying model representation texture images described by using a Markov random field. The results of the conducted experimental researches confirm that application of the developed approach allows to reduce the dimensionality of the features space while preserving the reliability of the classification. Texture analysis widespread in the processing of various types of images. However, despite the fact that even in 1979 Haralick noted that the methods of distinguishing textures are developed individually for each specific case [1], there is no clear definition of texture or a particular concept in solving problems analysis of texture images. Haindl wrote that the texture is a surface property, which is the spatial information contained in the object's surface [2]. The literature describes three approaches to texture analysis [1, 3, and 4]: A statistical approach, wherein the set of features used to provide texture image characteristics. Structural modeling allows us to consider texture as two-dimensional images composed of many primitives or subpatterns that are arranged accordance with a certain rule. Stochastic modeling suggests that the texture is the realization of a stochastic process that is characterized by certain parameters. This approach allows you to get good results for the generation of realistic natural texture images using Markov random fields [5]. To classification texture images, we will apply the model image as a realization of a random Markov field, that is, a stochastic approach to texture analysis. Great contribution to the development of this model has made by Haralick, who introduced the statistical and structural approaches to the description of texture [6] and suggested using of features based on the matrix of mutual probability distribution. The proposed is the gray level co-occurrence matrix [1]. It describes the spatial relationships of brightness pairs of texture elements.
1. Introduction
Introduction of stochastic models and random fields models have led to the development of image reconstruction algorithms,
segmentation, modeling and texture classification. In particular, Markov random fields is very useful for modeling spatial
relationships, as well as for the study of stochastic interaction between the observed values, including the analysis of medical
images and interpretation of remote sensing images [
The theory of Markov random field (MRF) provides a convenient and consistent method for modeling communication
between dependent entities, such as image pixels and correlated features. Convenience is achieved due to the characteristic
mutual influence among such objects, when using conditional distribution of MRF. The practical use of the model Markov
random field obtained thanks to the theorem of equivalence between MRF and the Gibbs distribution, which was introduced by
Hammersley and Clifford in 1971 [
We will consider the model of a Gaussian Markov random field (GMRF), which is a particular case of MRF, where the value
of the pixel in the position (i, j) is statistically independent of neighboring pixels. This means that the model takes into account
the spatial interaction between the various components within each color component, and interaction of [
Let ( , ) = [ 1( , ) 2( , ) … ( , )] is a vector in a texture region R. It is assumed that the vector at a position (i, j) represents the linear combination of the color components of neighboring pixels and additive Gaussian noise. Let 1, 2 … denote the mean color intensity, and 1, 2 … the spatial interaction of pixels and be the expected value of . x, y takes on the values from 1 to р. Let the associated parameters of the model and ∑ the co-occurrence matrix.
Spatial interaction of color pixels is defined as: Similarly it is defined for 2( , ), 3( , ) … ( , ). The generalized form is given by: ( , )∈ 12 −
∑ ( , )∈ 1 ( , )∈ 2 −
∑ ( , )∈
∑ ( , )∈ 11
∑ ( , )∈ 1 1( , ) = ( 1( , ) − 1) −
11( , )( 1( + , + ) − 1) − ∑ 12( , )( 2( + , + ) − 2) − ⋯
1 ( , )( ( + , + ) − ). ( , ) = ( ( , ) − ) −
1( , )( 1( + , + ) − 1) − ∑
2( , )( 2( + , + ) − 2) − ⋯ ( , )( ( + , + ) − ) , = ̅1̅̅,̅̅, where Nxy denote neighboring pixels. If х=у, then the neighboring pixels will correspond to the same color component. Otherwise, the neighboring pixels are of the other components.
The co-occurrence matrix is defined as follows: ∑ = ( 11 21
⋮ 1 2 12 22 1 2 ⋯ ⋱ ⋯ ⋮ ) . = [ ] = ∑ ( , )
( , ). 1 ( , )∈ The expected value is represented as: Having described all the terms, the probability density function of X(i,j) is found to be: ( ( , )| ) =
1 ((2 ) |Ʃ|)2 1 exp{ −1 2
( 1( , ) 2( , ) … ( , )) ∑( 1( , ) 2( , ) … ( , )) }.
neighborhood:
images on the textural classes.
criterion [
Winkler in "Image Analysis, Random Fields and Dynamic Monte Carlo Methods" [
Thus, in [
Individual criteria of separability for each feature were calculated (Figure 3).
For a sample consisting of n elements, divided into classes g and comprising a p features separability criterion is calculated a)
a) f x, y f x x, y y n
N = (( )−1 ),
= ∑ =1 ( − )(
− ), , = 1, , = ∑ =1 ∑ =1( − )( − ), , = 1, , using the following formulas: where =
+ . feature of class, = 1⁄
elements in class.
– is the intergroup dispersion matrix. The elements of this matrix are calculated according to the formula: − is the intragroup dispersion matrix. The elements of the matrix are calculated according to the formula: – is the value of the ˗ ℎ feature for the ˗ ℎ element of class, = 1 ⁄ ∑ =1 – is the mean value of the ˗ ℎ ∑ =1 – is the mean value of the ˗ ℎ feature in all the classes, and is the number of 0.74 0.58 0.58 0.53 0.77 0.74 0.61 0.82 0.85 0.77 0.61
0.55
Image Processing, Geoinformation Technology and Information Security / E. Biryukova, R. Paringer, A. Kupriyanov The higher the value of the criterion is, the more the separability of the classes grows.
After calculating the individual criteria for separability, features with a low value criterion were excluded. Analysis of the feature space, led to the conclusion that some of the neighboring pixels carry information about the features of the texture (pixel information are highlighted in Figure 3). It was excluded from the neighborhood of the pixels corresponding to non -informative features (calculated at (Δx = 2, Δy = 2), (Δx = -2, Δy = 1), (Δx = 1, Δy = 1), (Δx = -2, Δy = 0), (Δx = -1, Δy = 0), (Δx = -2, Δy = 1)). Thus, we resins are informative neighborhood new form. Modified neighborhood for the test classes is shown in Figure 4.
To study the effectiveness of the technology was evaluated the quality of the selected neighborhood. The evaluation was
conducted by calculating the clustering error on the based of k-means algorithm, where the centers of the starting classes used as
initial conditions [
Table 1 shows the values of the clustering error in the case of features, calculated using the causal neighborhood and modified to distinguish other classes of images from the selected base textures.
As Table 1 shows that clustering error value using the modified neighborhood does not exceed the error value using a causal neighborhood, which indicates the information content received surroundings, and hence the effectiveness of the proposed technology.
Figure 5 shows the mean values of separability criteria for the cases considered in Table 1, the modified neighborhoods are highlighted in color.
0.82 0.77 0.71 0.78 0.84 0.82 0.61 4. Conclusion 0.78 0.64 0.65 0.65 0.79 0.77 0.66 0.65 0.74 0.75 0.61 0.59 0.74 0.54 0.77 0.66 0.79 0.16 0.08 0.08 0.05 0.08 0.15 0.16 0.15 0.22 0.05 0.08 0.04 blanket1 and canvas1 scarf1 and scarf2 linseeds and sesameseeds
The paper presents the technology of choice informative neighborhood, which has shown to be effective for the considered classes of texture images. The features space and the clustering error were reduced by reducing the number of neighboring pixels. The proposed technique can be used to modeling texture images, wherein for the calculation of the model parameters using a Markov random field neighborhood.
Image Processing, Geoinformation Technology and Information Security / E. Biryukova, R. Paringer, A. Kupriyanov Acknowledgements
This work was partially supported by the Ministry of education and science of the Russian Federation in the framework of the implementation of the Program of increasing the competitiveness of SSAU among the world’s leading scientific and educational centers for 2013-2020 years; by the Russian Foundation for Basic Research grants (# 15-29-03823, # 15-29-07077, # 16-41630761, # 17-01-00972); by the ONIT RAS program # 6 “Bioinformatics, modern information technologies and mathematical methods in medicine” 2017.