Haralick Textural Features
These are a set of image features discovered by Wikipedia: Robert Haralick.
Algorithm
Texture vs Tone
Each image will have a tone and a texture:
- Tone - average color in a patch
- Texture - "variation of features of discrete gray tone"
Gray-Tone Spatial-Dependence matrices
Today, these are known as Gray Level Co-occurrence Matrix (GLCM).
For a matrix with \(\displaystyle N_g\), a Gray-Tone Spatial-Dependence matrix will be a \(\displaystyle N_g \times N_g\) symmetric matrix where entry \(\displaystyle i,j\) will contain the number of occurrences where a pixel with value \(\displaystyle i\) neighbors a pixel with value \(\displaystyle j\).
In an image each pixel will have eight neighboring pixels, except at the edges:
135° | 90° | 45° |
0° | * | 0° |
45° | 90° | 135° |
Then \(\displaystyle P(i,j,d,\alpha)\) is the number of occurrences where a pixel with value \(\displaystyle i\) and a pixel with value \(\displaystyle j\) are distance \(\displaystyle d\) apart along angle \(\displaystyle \alpha \in \{0^\circ, 45^\circ, 90^\circ, 135^\circ\}\).
If we fix d=1
, then we get four matrices of co-occurances along each direction:
- \(\displaystyle P_{H} = \{P(i,j,1, 0^\circ)\}\)
- \(\displaystyle P_{V} = \{P(i,j,1, 90^\circ)\}\)
- \(\displaystyle P_{LD} = \{P(i,j,1, 135^\circ)\}\)
- \(\displaystyle P_{RD} = \{P(i,j,1, 45^\circ)\}\)
For horizontal and vertical directions with resolution N, each row or column will have 2(N-1) neighbors. Thus in total, there will be \(\displaystyle R=2 N_x(N_y-1)\) or \(\displaystyle R=2 N_y(N_x-1)\) neighbors.
For diagonal directions, there will be \(\displaystyle R=2 (N_x - 1)(N_y-1)\) neighbors.
Each co-occurance matrix \(\displaystyle P\) can be normalized by dividing each entry by \(\displaystyle R\) to get \(\displaystyle p= P/R\).
Features
There are 14 values Haralick et al. compute per co-occurance matrix. The mean and range among the four matrices are used to get 28 features.
- Angular second moment: \(\displaystyle f_1 = \sum_i \sum_j p(i,j)^2\)
- Contrast:
- Correlation:
- Sum of squares variance:
- Inverse difference moment:
- Sum Average:
- Sum Entropy:
- Entropy:
- Difference Variance:
- Difference Entropy:
- Information Measures of Correlation 1:
- Information Measures of Correlation 2:
- Maximal Correlation Coefficient: