Texture Analysis

…uniformity, density, coarseness, roughness, regularity, intensity and directionality of discrete tonal features and their spatial relationship.

The textural analysis can be considered as one of applicable techniques for extracting image features. Texture is a measure of the variation of the intensity of a surface, quantifying properties such as smoothness, coarseness and regularity. Texture has been one of the most important characteristics which have been used to classify and recognize objects and scenes. It can be characterized by textural primitives as unit elements and neighborhoods in which the organization and relationships between the properties of these primitives are defined. Haralick and Shapiro (1991) defined texture as the uniformity, density, coarseness, roughness, regularity, intensity and directionality of discrete tonal features and their spatial relationships. Although no generally applicable definition of texture exists, some common elements in the definitions found in the literature are primitives and/or properties that are defined in a neighborhood and the statistical and/or structural relationships between these primitives and/or properties that are measured at a scale of interest.
Structural approaches have been one of the major research directions for texture analysis. They use the idea that texture is composed of primitives with different properties appearing in particular spatial arrangements. On the other hand, statistical approaches try to model texture using statistical distributions either in the spatial domain or in a transform domain. One way to combine these two approaches is to define texture as being specified by the statistical distribution of the properties of different textural primitives occurring at different spatial relationships. A pixel, with its gray level as its property, is the simplest primitive that can be defined in a digital image. Consequently, distribution of pixel gray levels can be described by first-order statistics like mean, standard variation, skewness and kurtosis or second-order statistics like the probability of two pixels having particular gray levels occurring at particular spatial relationships. This information can be summarized in two-dimensional co-occurrence matrices computed for different distances and orientations. Coarse textures are ones for which the distribution changes slightly with distance, whereas for fine textures the distribution changes rapidly with distance.

Two-dimensional co-occurrence (gray-level dependence) matrices are generally used in texture analysis because they are able to capture the spatial dependence of gray-level values within an image. Co-occurrence, in general form, can be specified in a matrix of relative frequencies P(i,j;d,θ) with which two neighboring texture elements separated by distance d at orientation θ occur in the image, one with property i and the other with property j. In gray level co-occurrence, as a special case, texture elements are pixels and properties are gray levels. For example, for a 0o angular relationship, P(i,j;d,0o) averages the probability of a left-right transition of gray level i to gray level j at a distance d.
In the derivations below as described in equation (1), origin of the image is defined as the upper-left corner pixel. Let Lr = {0,1,…,Nr-1} and Lc = {0,1,…,Nc-1} be the spatial domains of row and column dimensions, and G = {0,1,…,Ng-1} be the domain of gray levels. The image I can be represented as a function as a function which assigns a gray level to each pixel in the domain of the image; I: Lr x Lc →G. Then, for the orientations shown in figure 1, gray level co-occurrence matrices can be defined as:
P(i,j;d,0o)=#{((r,c),(r’,c’))∊ (Lr x Lc)x(Lr x Lc)| r’-r=0,|c’-c|=d, I(r,c)=i, I(r’,c’)=j}
P(i,j;d,45o)=#{((r,c),(r’,c’))∊ (Lr x Lc)x(Lr x Lc)|
(r’-r=d, c’-c=d), or (r’-r=-d, c’-c=-d, )I(r,c)=i, I(r’,c’)=j}
P(i,j;d,90o)=#{((r,c),(r’,c’))∊ (Lr x Lc)x(Lr x Lc)| |r’-r|=d,c’-c=0, I(r,c)=i, I(r’,c’)=j}
P(i,j;d,130o)=#{((r,c),(r’,c’))∊ (Lr x Lc)x(Lr x Lc)|
(r’-r=d, c’-c=-d), or (r’-r=-d, c’-c=d, )I(r,c)=i, I(r’,c’)=j}                      (1)

Fig. 1. Spatial arrangements of pixels.

Resulting matrices are symmetric. The distance metric used in equation (2) can be explicitly defined as:
ρ((r,c),(r’,c’))=max{|r-r’|,|c-c’|}                                               (2)
These matrices can be normalized by dividing each entry in a matrix by the number of neighboring pixels in computing that matrix. Given distance d, this number is 2Nr(Nc-d) for 0 orientation, 2(Nr-d)(Nc-d) for 45 and 135 orientations, and 2(Nr-d)Nc for 90 orientation.

The HSV and HSL CCM texture analysis method consists of: (1) transformation of images from RGB color space to HSV and HSL color space; (2) generation of spatial grey-level dependence matrices (SGDMs); and (3) determination of Haralick textural features. The GLCM involves conversion of RGB images to gray-level images before extraction of the SGDM with the equation as following:

Grey = 0.3R + 0.59G + 0.11B                                       (3)

The range of RGB CCM, HSV CCM, HSL CCM and GLCM within a given image determines the dimensions of a two-dimensional co-occurrence matrix. In each of RGB CCM, HSV CCM, HSL CCM and GLCM have 256 grey-levels, which would make the co-occurrence matrix 256 x 256.

Ten textural features used in this research can be described as following:

where: P(i,j) is the (i,j) element of a normalized co-occurrence matrix, and μ　and σ are the mean and standard deviation of the pixel element given by the following relationships:

where: N(i,j) is the number counts in the image with pixel intensity i followed by pixel intensity j at one pixel displacement to the left, and M is the total number of pixels.

These are the definition of each textural features:

Entropy measures the randomness of a gray-level distribution. The entropy is expected to be high if the grey levels are distributed randomly through out the image.

Energy measures the number of repeated pairs. The energy is expected to be high if the occurrence of repeated pixel pairs is high.

Contrast measures the local contrast image. The contrast is expected to be low if the gray levels of each pixel pair are similar.

Homogeneity measures the local homogeneity of a pixel pair. The homogeneity is expected to be large if the gray levels of each pixel pair are similar.

Sum Mean provides the mean of the gray levels in the image. The Sum Mean is expected to be large if the sum of the gray levels of the image is high.

Variance tells us how spread out the distribution of gray levels is. The variance is expected to be large if the gray levels of the image are spread out greatly.

Correlation provides a correlation between the two pixels in the pixel pair. The correlation is expected to be high if the gray levels of the pixel pairs are highly correlated.

Maximum Probability is the results in the pixel pair that is most predominant in the image. The maximum probability is expected to be high if the occurrence of the most predominant pixel pairs is high.

Inverse Different Moment tells us about the smoothness of the image, like homogeneity. The inverse different moment is expected to be high if the gray levels of the pixel pairs are similar.

Cluster Tendency measures the grouping of pixels that have similar gray level values.