Haralick Textural Features: Difference between revisions

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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>.

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:

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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>.

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 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>

# 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:

#* Note that the original paper has a typo.

# Difference Variance:

# Entropy: <math>f_9 = - \sum_i \sum_j p(i,j) \log(p(i,j))</math>

# Difference Entropy:

# 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:

# 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==

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 |}
-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>.
+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:
@@ Line 33: / Line 33: @@
 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>.
+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 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>
-# 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:
+#* Note that the original paper has a typo.
-# Difference Variance:
+# Entropy: <math>f_9 = - \sum_i \sum_j p(i,j) \log(p(i,j))</math>
-# Difference Entropy:
+# 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:
+# 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==