Statistical Methods of Texture Analysis

• Concept: Texture analysis often relies on spatial relationships between pixel gray levels, not just individual pixel values.
• Gray-Level Co-occurrence Matrix (GLCM):
  • Defined by a displacement vector $\mathbf{d}=(dx,dy)$, representing the spatial relationship between pixel pairs.
  • $P[i,j]$ counts the number of pixel pairs separated by $\mathbf{d}$ with gray levels $i$ and $j$.
  • Normalized $P[i,j]$ can be treated as a probability mass function. Okay, here’s a breakdown of the Texture Features derived from the Gray-Level Co-occurrence Matrix (GLCM), designed for better understanding:

What is the GLCM ?

• The GLCM captures how often different combinations of pixel gray levels occur next to each other in an image, based on a defined spatial relationship (the displacement vector d).
• Think of it as a matrix that summarizes the spatial distribution of gray levels in a textured region.

Texture Features from the GLCM: What do they tell us?

These features quantify different aspects of the texture based on the probabilities in the GLCM. They help us distinguish between different types of textures.

1. Entropy:

   • What it measures: The randomness or disorder in the texture.
   • Intuition:
     • High entropy = Many different gray-level pairs occur with similar probability (like a noisy or very complex texture). The GLCM will have values spread out.
     • Low entropy = Only a few gray-level pairs dominate (like a very regular or uniform texture). The GLCM will have a few large values.
   • Formula: $$\text{Entropy}=-\sum _i\sum _{j}P[i,j]\log P[i,j]$$
     • $P[i,j]$ is the probability of gray levels $i$ and $j$ occurring together (from the GLCM).

2. Energy (also called Angular Second Moment):

   • What it measures: The uniformity or homogeneity of the texture.
   • Intuition:
     • High energy = The texture is very uniform (only a few gray-level pairs are common). The GLCM will have a few large values, possibly close to the diagonal if the pixel pairs often have similar gray level values.
     • Low energy = The texture is less uniform, with many different gray-level pairs present.
   • Formula: $$\text{Energy}=\sum _i\sum _j{P}^2[i,j]$$
     • We square the probabilities, so larger $P[i,j]$ values contribute much more to the sum.

3. Contrast:

   • What it measures: The amount of local variation in gray levels in the texture.
   • Intuition:
     • High contrast = Large differences between gray levels of neighboring pixels are common (like a texture with sharp edges or large variations between light and dark areas). Values further away from the diagonal of the GLCM will be larger.
     • Low contrast = Neighboring pixels tend to have similar gray levels (like a smooth texture). Values along the diagonal will be larger.
   • Formula: $$\text{Contrast}=\sum _i\sum _j(i-j{)}^{2}P[i,j]$$
     • The $(i-j{)}^2$ term emphasizes large differences in gray levels.

4. Homogeneity:

   • What it measures: How close the distribution of elements in the GLCM is to the GLCM diagonal. This indicates how similar the pixel pairs are in terms of their gray level values.
   • Intuition:
     • High homogeneity = Pixel pairs tend to have very similar gray level values (the texture is locally smooth). Values are concentrated along the diagonal of the GLCM.
     • Low homogeneity = Pixel pairs often have different gray level values (the texture has more local variations).
   • Formula: $$\text{Homogeneity}=\sum _i\sum _j\frac{P[i,j]}{1+|i-j|}$$
     • The $1+|i-j|$ term in the denominator gives more weight to pixel pairs with similar gray levels (where $|i-j|$ is small).

In Summary

• These four features (Entropy, Energy, Contrast, and Homogeneity) provide a statistical summary of texture characteristics based on the GLCM.
• They are often used together in texture classification and segmentation tasks.
• The specific values of these features will depend on the choice of the displacement vector d used to create the GLCM.

I hope this more detailed explanation makes these texture features easier to understand!

Page 6: Structural Analysis & Autocorrelation

• Structural Analysis of Ordered Texture: Used when texture primitives are large and can be individually segmented.
  • Involves describing primitives (e.g., shape, size) and their spatial arrangement (placement rules).
  • Morphological methods can be helpful for extracting primitives in noisy images.
• Autocorrelation: Measures self-similarity of an image at different spatial lags.
  • For an $N\times N$ image, the autocorrelation function $p[k,l]$ is defined as: $$p[k,l]=\frac{{\sum }_{i=1}^{(N-k)}{\sum }_{j=1}^{(N-l)}f[i,j]f[i+k,j+l]}{{\sum }_{i=1}^N{\sum }_{j=1}^N{f}^2[i,j]},0\le k,l\le N-1(7.5)$$ where:
    • $f[i,j]$ is the pixel value at location $(i,j)$.
    • $k$ and $l$ are spatial lags.
  • Periodic textures show periodic behavior in the autocorrelation function.
  • Coarse textures: autocorrelation drops off slowly.
  • Fine textures: autocorrelation drops off rapidly.