Haralick Textural Features: Difference between revisions
Line 38: | Line 38: | ||
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. | 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: <math>f_1 = \sum_i \sum_j p(i,j)^2</math> | # Angular second moment: <math>f_1 = \sum_i \sum_j p(i,j)^2</math> | ||
# Contrast: | # Contrast: <math>f_2 = \sum_{n=0}^{N_{g-1}} n^2 \{ \underset{|i-j|=n}{\sum_{i=1}^{N-g}\sum_{j=1}^{N-g}} p(i,j) \}</math> | ||
# Correlation: | # Correlation: <math>f_3 = \frac{\sum_i \sum_j (ij)p(i,j) - \mu_x \mu_y}{\sigma_x \sigma_y}</math> | ||
# Sum of squares variance: | # Sum of squares variance: <math>f_4 = \sum_i \sum_j (i-\mu)^2 p(i,j)</math> | ||
# Inverse difference moment: | # Inverse difference moment: <math>f_5 = \sum_i \sum_j \frac{1}{1+(i-j)^2} p(i,j)</math> | ||
# Sum Average: | # Sum Average: <math>f_6 = \sum_{i=2}^{2N_g} ip_{x+y}(i)</math> | ||
# Sum Entropy: | # Sum Entropy: <math>f_7 = \sum_{i=2}^{2N_g} (i-f_6)^2 p_{x+y}(i)</math> | ||
# Entropy: | # Entropy: | ||
# Difference Variance: | # Difference Variance: |
Revision as of 14:39, 3 May 2022
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: \(\displaystyle f_2 = \sum_{n=0}^{N_{g-1}} n^2 \{ \underset{|i-j|=n}{\sum_{i=1}^{N-g}\sum_{j=1}^{N-g}} p(i,j) \}\)
- Correlation: \(\displaystyle f_3 = \frac{\sum_i \sum_j (ij)p(i,j) - \mu_x \mu_y}{\sigma_x \sigma_y}\)
- Sum of squares variance: \(\displaystyle f_4 = \sum_i \sum_j (i-\mu)^2 p(i,j)\)
- Inverse difference moment: \(\displaystyle f_5 = \sum_i \sum_j \frac{1}{1+(i-j)^2} p(i,j)\)
- Sum Average: \(\displaystyle f_6 = \sum_{i=2}^{2N_g} ip_{x+y}(i)\)
- Sum Entropy: \(\displaystyle f_7 = \sum_{i=2}^{2N_g} (i-f_6)^2 p_{x+y}(i)\)
- Entropy:
- Difference Variance:
- Difference Entropy:
- Information Measures of Correlation 1:
- Information Measures of Correlation 2:
- Maximal Correlation Coefficient: