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CHAPTER 7 TEXTURE
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o 1 J
P[i, j] for d = (1,0)
P[i, j] for d =
(1,1)
Figure
matrix
72: (a) An for d
checkerboard
image
(b) The
gray-level co-occurrence matrix for d
(1,1) (c) The
gray-level co-occurrence
(1,0)
will not have any preferred set of gray-level pairs In such a case the matrix is expected to be uniformly populated Thus, a feature which measures the randomness of gray-level distribution is the entropy, defined as Entropy = -
L L P[i,j] logP[i,j] i
(71)
Note that the entropy is highest when all entries in P[i, j] are equal; such a matrix corresponds to an image in which there are no preferred gray-level
pairs for the specifieddistance vector d The features of
and homogeneity are
energy, contrast,
also defined using the gray-level co-occurrence matrix Energy =
Contrast
as given below:
L L p2 [i,j]
(72) (73) (74)
= L L(i
- j)2 P[i, j] P[i,j]
Homogeneity=
2;= 2;= 1 t J
+ Ii - J
The choice of the displacement vector d is an important parameter in the definition of the gray-level co-occurrence matrix Occasionally, the co-occurrence
73STRUCTURAL
ANALYSIS OF ORDERED TEXTURE
matrix is computed for several values of d and the one which maximizes a statistical measure computed from P[i, j] is used The gray-level co-occurrence matrix approach is particularly suitable for describing microtextures It is not suitable for textures comprising large area primitives since it does not capture shape properties Gray-level co-occurrence matrices have been used extensively in remote sensing applications for land-use classification Autocorrelation The autocorrelation function p[k, I] for an N x N image is defined as follows: 1--- ,,(N-k) (N-l) p[k, I] = (N-k)(N-l) L,i=l Lj=l f[~,J] f[~+k,j+l]
J2 Lf:l Lf=l P[i,l
' 0 ~ k, I ~ N-l (75)
For images comprising repetitive texture patterns the autocorrelation function exhibits periodic behavior with a period equal to the spacing between adjacent texture primitives When the texture is coarse, the autocorrelation function drops off slowly, whereas for fine textures it drops off rapidly The autocorrelation function is used as a measure of periodicity of texture as well as a measure of the scale of the texture primitives
Structural Analysis of Ordered Texture
When the texture primitive is large enough to be individually segmented and described, then structural methods which describe the primitives and their placement rules are useful For example, consider a simple texture formed by the repeated placement of homogeneous gray-level discs in a regular grid pattern as shown in Figure 73(a) Such a texture can be described by first segmenting the discs using a simple method such as connected component labeling, described earlier, and then determining the regular structure formed by the centroids of these connected components For more general binary images the primitives can be first extracted using morphological methods and then their placement rules determined Such morphological methods are particularly useful when the image is corrupted by noise or other nonrepeating random patterns which would be difficult to separate in a simple connected component method For example, when the image shown in Figure 73(a) is
CHAPTER
TEXTURE
Figure
73: (a) A simple texture formed by repeated placement of discs on a
regular grid (b) Texture in (a) corrupted by random streaks of lines corrupted by noise resulting in random streaks of lines as shown in Figure 73(b), morphological techniques (see 2) can be used to locate all discs For gray-scale images we can define a predicate which is satisfied by all pixels within each blob corresponding to a primitive A commonly used predicate is the gray-level homogeneity predicate The image is initially processed using a Laplacian of Gaussian filter (see 5) Primitive regions are then identified by grouping all those pixels which are not on or near edge pixels For homogeneous blobs properties such as size, elongation, and orientation are useful features Measures based on co-occurrence of these primitives obtained by analyzing their spatial relationship are then used to characterize the texture
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