Haralick Textural Features: Difference between revisions

Revision as of 15:12, 2 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:

Nearest Neighbors to *
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 R: \(\displaystyle p= P/R\).

Features

There are 14 values Haralick et al. compute per co-occurance matrix. The mean and range among the 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:

@@ 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>.
+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 R: <math> p= P/R</math>.
+===Features===
+There are 14 values Haralick ''et al.'' compute per co-occurance matrix. The mean and range among the matrices are used to get 28 features.
+# Angular second moment: <math>f_1 = \sum_i \sum_j p(i,j)^2</math>
+# 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:
 ==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]
+* [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]