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Texture analysis is important in several image segmentation and classification problems. Different image textures manifest themselves by dissimilarity in both the property values and the spatial interrelationships of their component texture primitives. We use this fact in a texture discrimination system. This paper focuses on how to apply texture operators based on co-occurrence matrix, texture filters and fractal dimension to the problem of object recognition and image segmentation.

Unsupervised image segmentation is a fundamental issue in image analysis and computer vision. The purpose of segmentation is to partition the image into regions of similar attribute like luminance, color or texture. Texture plays an important role in numerous computer vision applications, particularity in segmentation of images. Many useful properties for image description and interpretation are gained through texture observation and analysis. Texture classification categories are based sometimes on distinguishing feature as shown in [

On the other hand, there are some structural methods for texture analysis that can be thought as the regular repetition of a micro texture and the fractal dimension may be appropriate for their characterization.

The aim of this report is to compare the effectiveness of some segmentation methods based on co-occurrence matrices, moreover the texture filters introduced by [

The algorithms described in this paper are unsupervised in the following sense:

• They do not require a prior knowledge on the textures in the image, in particular no learning step is necessary;

• The number or regions or texture classes needs not to be known either.

The techniques discussed are neighborhood operators that access pixels in a small area around each central pixel and derive a new value for it, performing some calculation with those values. The operator is repeated throughout the image and the values obtained produce a new image.

The paper is organized as follows, the co-occurrence matrix is introduced in Section 2, the Texture filters in Section 3 and the Method of Range to calculate fractal dimension in Section 4. The performance of each method is illustrated in each section. Conclusions are presented in Section 5.

The gray level co-occurrence matrix is defined as a joint distribution of the gray levels of two pixels separated by a given displacement τ . In Cartesian coordinates the displacement can be chosen as a vector τ = ( Δ x , Δ y ) . Each element in the matrix, A τ ( g , g ′ ) , is defined as

A τ ( g , g ′ ) = { ( g , g ′ ) | i m a g e ( x , y ) = g ∧ i m a g e ( x + Δ x , y + Δ y ) = g ′ }

where g and g ′ are gray levels in the image. The size of the co-occurrence matrix A τ is G × G , denoting G the number of gray levels in the image. This concept is illustrated with a binary model for τ = ( 0 , 1 ) in

Haralick et al. [

∑ g , g ′ A τ ( g , g ′ ) 2 (1)

This feature is also called angular second moment and measures textures uniformity, that is, pixel pairs repetition. When the image is homogenous, this is only similar gray level pixels are present, energy reaches its maxi-mum. Thus, high values of energy occur when the gray level distribution over the window has either a constant or a periodic form.

Entropy measures the disorder of an image. It assumes larger values when the image is not texturally uniform. It is important to remark that conceptually energy and entropy are inversely correlated.

Local homogeneity, also called inverse difference moment, measures images homogeneity. It assumes larger values for smaller gray tone difference in pair elements. The correlation between the local homogeneity and the energy is very strong but in an inverse way.

Correlation is a measure of gray level linear dependencies in the image and high correlation values imply a linear relationship between the gray levels of pixel pair. Thus the correlation is uncorrelated to energy and entropy, that is, to pixel pair repetitions.

Several co-occurrence matrices in different directions, for example, 0˚, 45˚, 90˚, and 135˚, this is for different displacement

that can be interpreted as a normalization of the scene energy to the gray level linear dependence in the image. Some of these operations are obviously easier than others to calculate for all the pixels in an image. The resulting values are scaled to create a new image that can be segmented by brightness threshold. Sometimes experimentation with several texture operators is required to find the one that gives the best separation be-tween the object and its surrounding and in consequence the best segmentation. The results of applying some of these operators are illustrated in

The resulting texture characteristics are often redundant and too numerous, particularly when using more displacement vectors. To overcome these drawbacks, in the 90’s approach texture spectrum, based on texture units characterizing local texture information in the eight directions proposed. This method has been applied to the extraction of texture characteristics, classification of textures, detection of edges and filtering of textures [

A statistical method for texture analysis focuses on the characterization and discrimination of textures was presented by He and Wang in [

Each pixel is surrounded by its eight nearest neighbors of order two that represents the smallest complete unit, in the sense that it has the eight directions around the pixel, and where you can extract the information for the local texture. For each pixel s take a

and the element

Since each element of texture unit can take three possible values, the total amount of texture units are

There is no a single way to label and sort the 6561 texture units, He and Wang [

that represents the “number of texture unit”, with

The 6561 set of texture units describes the local texture appearance of a given pixel, this is the relative relationship between central gray levels and its neighbors and the frequency of occurrence of all the texture units over the image will reveal information on texture. We call “texture spectrum” to the frequency distribution of all texture units. Graphically, the horizontal axis represent the number of texture units, N_{UT}, and the ordinate axis represent frequency of occurrence.

Texture spectrum in the increased percentage of a texture component will result in a tendency to a particular distribution of peaks. And the different textures are composed of texture units in individuals with different distributions textured appearance, thus the texture of an image can be characterized by its texture spectrum. Let us note that although the labeling method chosen can affect the relative position of the texture units in the spectrum of texture, not change the value of its frequency in the spectrum. We also make clear that the local texture of a pixel and its environment is characterized by the corresponding texture unit, while the appearance of the texture reveals its texture spectrum calculated and a suitable window size not depends on window the nature of the image.

Al-Jacobi, in [

Each element of these new units can take the values 0, 1 or 2 in the same manner as in (6). Now these texture units can take

Bhattacharya and others proposed in [

where

After estimating the

Unit Texture Algorithm

• A centered squared

• The texture number

• For each value of

− Mean filter: every pixel having the same

− Median filter: every pixel having the same

In

Let

in other words

Several methods are use to calculate the fractal dimension, we propose the Method of Range.

The Method of Range, which has been introduced in [

of the brightness range, keeping the idea of refining the grid over the image, while improving the efficiency of the computation [

• A centered squared window of size

• For the same pixel, we take a small squared window of size

• We estimate the fractal dimension as

• The histogram with all the values D is constructed and, choosing a threshold h, the image is recolored putting black on each pixel where

The expression D represents the ratio between the difference of the ranges in each window and the proportion of the length of each window, in log scale. This segmentation technique is illustrated in

One of the more important indication of the usefulness of an unsupervised image segmentation algorithm, is its stability. We need segmentation with low bias and variance; this is low variability respect to the parameters.

In this paper each method was applied to the segmentation of the same images to measure the strengths and weaknesses, not in the images where they have their best performance.

About the texture features based on co-occurrence matrices, energy, entropy, local homogeneity and correlation were chosen. All of them were computed for different values of

In that sense, the algorithms based on Unit Texture provide a profile in the image improving a bit the separation of regions of different texture and with less computational cost. In most of the images in which we have applied them, both algorithms produce very similar results; however, the medium filter is slightly more effective in reducing noise

The Range Method, based on the fractal dimension, has also been effective in most of the images and is computationally faster. The lower computational weight is not the only advantage of the method, in addition to this method defined the edges very well, and the segmentation was not so homogeneous in the large regions; it not only does profile the coast well but it also detects objects inside.

The Range Method has been shown to be an improvement in statistical methods based on MRF and could be used to build a supervised classifier based on previous segmented images as training blocks with low computational cost.

The authors declare no conflicts of interest regarding the publication of this paper.

Marrón, B. (2018) Texture Filters and Fractal Dimension on Image Segmentation. Journal of Signal and Information Processing, 9, 229-238. https://doi.org/10.4236/jsip.2018.93014