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Equations for 3D Haralick Texture features[1], which are based in a gray-level co-occurrence matrix of the image
==Notation==
 

   --<math>p(i,j)</math>: (i,j)th entry in a normalized gray-tone spatial dependence matrix, <math>p(i,j)= P(i,j)/R</math>
     * <math>P(i,j)</math> is the co-occurrence matrix and ''R'' is the sum of values in it, thus <math>P(i,j)</math> can be  considered as the joint distribution 
     of ''i'' and ''j'', which are gray levels of the original image. The value of entry p(i,j) is supposed to be very small due to the 
     large size of the co-occurrence matrix.

   --<math>p_x(i)/p_y(i)</math>: ''i''th entry in the marginal-probability distribution matrix obtained by summing the rows/columns of <math>p(i,j)</math>.

   --<math>N_g</math>: Number of distinct gray levels in the image.

    
   --<math>p_{x+y}(k):</math> <math> p_{x+y}(k) = \sum_{i=1}^{N_g} \sum_{j=1,i+j=k}^{N_g} p(i,j), k=2,3,...,2N_g </math>
     <math>p_{x+y}(i)</math> is the probability of co-occurrence matrix coordinates summing to x+y

   --<math>p_{x-y}(k):</math> <math> p_{x-y}(k) = \sum_{i=1}^{N_g} \sum_{j=1,|i-j|=k}^{N_g} p(i,j), k=0,1,...,N_g-1 </math>


==Textural Features==
* 1) Angular Second Moment:<math>f_1 =  \sum_{i=1}^{N_g}  \sum_{j=1}^{N_g} p(i,j)^2 </math>
* 2) Contrast:<math>f_2 =  \sum_{n=0}^{N_g - 1} n^2 {\sum_{i=1}^{N_g} \sum_{j=1,|i-j|=n}^{N_g} p(i,j)} </math>
* 3) Correlation:<math>f_3 = \cfrac{\sum_i \sum_j (i,j)p(i,j)-u_x u_y}{\sigma_x \sigma_y}</math>, where <math>u_x</math>,<math>u_y</math>,<math>\sigma_x</math>,<math>\sigma_y</math> are the means and std.deviations of  <math>p_x</math> and <math>p_y</math>, the partial probability density functions
* 4) Sum of the Squares of Variance: <math>f_4 = \sum_{i=1}^{N_g} \sum_{j=1}^{N_g} (i-j)^2 p(i,j)</math>
* 5) Inverse Difference Moment: <math>f_5 = \sum_i \sum_j \cfrac{1}{1+(i-j)^2}p(i,j)</math>
* 6) Sum Average: <math>f_6 = \sum_{i=2}^{2N_g} ip_{x+y}(i)</math>
* 7) Sum Variance: <math>f_7 = \sum_{i=2}^{2N_g}(i-f_8)^2 p_{x+y}(i)</math>
* 8) Sum Entropy: <math>f_8 = - \sum_{i=2}^{2N_g}p_{x+y}(i)log(p_{x+y}(i))</math>
* 9) Entropy: <math>f_9 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p(i,j))</math>
* 10) Difference Variance: <math>f_{10} = \sum_{i=0}^{N_g-1} i^2 p_{x-y}(i)</math>
* 11) Difference Entropy: <math>f_{11}= -\sum_{i=0}^{N_g-1} p_{x-y}(i)log(p_{x-y}(i))</math>
* 12) Information Measures of Correlation 1: <math>f_{12} = \cfrac{HXY-HXY1}{max(HX,HY)}</math>
* 13) Information Measures of Correlation 2: <math>f_{13} = (1-exp(-2.0|HXY2-HXY|))^{1/2}</math>, with <math>HXY = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p(i,j))</math>,
<math>HXY1 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p_x(i) p_y(i))</math>,
<math>HXY2 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p_x(i)p_y(j)log(p_x(i) p_y(i))</math>

For more information about the Textural features, please refer to [1].
== Reference ==
* [1] R. Haralick, K. Shanmugam, and I. Dinstein, "Textural features for image classification," IEEE Transactions on Systems, Man, and Cybernetics, SMC-3, 610-621, 1973.