Haralick Textural Features: Difference between revisions
Created page with " ==Resources== * [https://ieeexplore.ieee.org/document/4309314 Textural Features for Image Classification (1973)] [https://www.researchgate.net/publication/302341151_Textural_Features_for_Image_Classification Researchgate Mirror]" |
|||
(10 intermediate revisions by the same user not shown) | |||
Line 1: | Line 1: | ||
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).<br> | |||
For a matrix with <math>N_g</math>, a Gray-Tone Spatial-Dependence matrix will be a <math>N_g \times N_g</math> symmetric matrix where entry <math>i,j</math> will contain the number of occurrences where a pixel with value <math>i</math> neighbors a pixel with value <math>j</math>.<br> | |||
In an image each pixel will have eight neighboring pixels, except at the edges: | |||
{| class="wikitable" style="margin:auto;text-align:center" | |||
|+ Nearest Neighbors to * | |||
|- | |||
| 135° || 90° || 45° | |||
|- | |||
| 0° || * || 0° | |||
|- | |||
| 45° || 90° || 135° | |||
|} | |||
Then <math>P(i,j,d,\alpha)</math> is the number of occurrences where a pixel with value <math>i</math> and a pixel with value <math>j</math> are distance <math>d</math> apart along angle <math>\alpha \in \{0^\circ, 45^\circ, 90^\circ, 135^\circ\}</math>. Note that each neighbor pair is counted twice (e.g. pixel 1 is neighbor of 0 and 0 is neighbor of 1). | |||
If we fix <code>d=1</code>, then we get four matrices of co-occurances along each direction: | |||
* <math>P_{H} = \{P(i,j,1, 0^\circ)\}</math> | |||
* <math>P_{V} = \{P(i,j,1, 90^\circ)\}</math> | |||
* <math>P_{LD} = \{P(i,j,1, 135^\circ)\}</math> | |||
* <math>P_{RD} = \{P(i,j,1, 45^\circ)\}</math> | |||
For horizontal and vertical directions with resolution N, each row or column will have 2(N-1) neighbors. Thus in total, there will be <math>R=2 N_x(N_y-1)</math> or <math>R=2 N_y(N_x-1)</math> neighbors.<br> | |||
For diagonal directions, there will be <math>R=2 (N_x - 1)(N_y-1)</math> neighbors.<br> | |||
Each co-occurance matrix <math>P</math> can be normalized by dividing each entry by <math>R</math> to get <math> p= P/R</math>. | |||
===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> | |||
# 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: <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: <math>f_4 = \sum_i \sum_j (i-\mu)^2 p(i,j)</math> | |||
# Inverse difference moment: <math>f_5 = \sum_i \sum_j \frac{1}{1+(i-j)^2} p(i,j)</math> | |||
# Sum Average: <math>f_6 = \sum_{i=2}^{2N_g} ip_{x+y}(i)</math> | |||
# Sum Entropy: <math>f_7 = \sum_{i=2}^{2N_g} (i-f_6)^2 p_{x+y}(i)</math> | |||
#* Note that the original paper has a typo. | |||
# Entropy: <math>f_9 = - \sum_i \sum_j p(i,j) \log(p(i,j))</math> | |||
# Difference Variance: <math>f_{10}= var(p_{x-y})</math> | |||
# Difference Entropy: <math>f_{11} = -\sum_{i=0}^{N_{g}-1} p_{x-y}(i) \log p_{x-y}(i)</math> | |||
# Information Measures of Correlation 1: | |||
# Information Measures of Correlation 2: | |||
# Maximal Correlation Coefficient: <math>f_{14} = (second largest eigenvalue of Q)^{1/2}</math> where <math>Q(i,j) = \sum_k \frac{p(i,k)p(j,k)}{p_x(i)p_y(k)}</math> | |||
Notation: | |||
* <math>p(i,j)</math> = i,j value in the noramlized co-occurance matrix | |||
* <math>p_x(i)</math> = marginal probability (<math>\sum_j p(i,j)</math>) | |||
* <math>N_g</math> number of gray tones | |||
==Resources== | ==Resources== | ||
* [https://ieeexplore.ieee.org/document/4309314 Textural Features for Image Classification (1973)] [https://www.researchgate.net/publication/302341151_Textural_Features_for_Image_Classification | * [https://ieeexplore.ieee.org/document/4309314 Textural Features for Image Classification (1973)] | ||
** [http://haralick.org/journals/TexturalFeatures.pdf Mirror] [https://www.researchgate.net/publication/302341151_Textural_Features_for_Image_Classification ResearchGate] | |||
* [https://murphylab.web.cmu.edu/publications/boland/boland_node26.html Haralick texture features explanation] | |||
===Implementations=== | |||
* [https://github.com/sentinel-hub/eo-learn/blob/master/features/eolearn/features/haralick.py eo-learn] | |||
* [https://github.com/DigitalSlideArchive/HistomicsTK/blob/master/histomicstk/features/compute_haralick_features.py HistomonicsTK] |
Latest revision as of 14:47, 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 \(\displaystyle \alpha \in \{0^\circ, 45^\circ, 90^\circ, 135^\circ\}\). Note that each neighbor pair is counted twice (e.g. pixel 1 is neighbor of 0 and 0 is neighbor of 1).
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: \(\displaystyle f_7 = \sum_{i=2}^{2N_g} (i-f_6)^2 p_{x+y}(i)\)
- Note that the original paper has a typo.
- Entropy: \(\displaystyle f_9 = - \sum_i \sum_j p(i,j) \log(p(i,j))\)
- Difference Variance: \(\displaystyle f_{10}= var(p_{x-y})\)
- Difference Entropy: \(\displaystyle f_{11} = -\sum_{i=0}^{N_{g}-1} p_{x-y}(i) \log p_{x-y}(i)\)
- Information Measures of Correlation 1:
- Information Measures of Correlation 2:
- Maximal Correlation Coefficient: \(\displaystyle f_{14} = (second largest eigenvalue of Q)^{1/2}\) where \(\displaystyle Q(i,j) = \sum_k \frac{p(i,k)p(j,k)}{p_x(i)p_y(k)}\)
Notation:
- \(\displaystyle p(i,j)\) = i,j value in the noramlized co-occurance matrix
- \(\displaystyle p_x(i)\) = marginal probability (\(\displaystyle \sum_j p(i,j)\))
- \(\displaystyle N_g\) number of gray tones