View source for 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).<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==
* [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]