Equations for 3D Haralick Texture Feature

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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==
 
==Notation==
 
   
 
   
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     large size of the co-occurrence matrix.
 
     large size of the co-occurrence matrix.
  
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    --<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>.
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  --<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>.
  
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    --<math>N_g</math>: Number of distinct gray levels in the image.
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  --<math>N_g</math>: Number of distinct gray levels in the image.
  
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     --<math>p_{x+y}(k)</math>: <math> p_{x+y}(k) = \sum_i x</math>:
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  --<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>
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    <math>p_{x+y}(i)</math> is the probability of co-occurrence matrix coordinates summing to x+y
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  --<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>
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==Textural Features==
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* 1) Angular Second Moment:<math>f_1 =  \sum_{i=1}^{N_g}  \sum_{j=1}^{N_g} p(i,j)^2 </math>
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* 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>
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* 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
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* 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>
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* 5) Inverse Difference Moment: <math>f_5 = \sum_i \sum_j \cfrac{1}{1+(i-j)^2}p(i,j)</math>
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* 6) Sum Average: <math>f_6 = \sum_{i=2}^{2N_g} ip_{x+y}(i)</math>
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* 7) Sum Variance: <math>f_7 = \sum_{i=2}^{2N_g}(i-f_8)^2 p_{x+y}(i)</math>
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* 8) Sum Entropy: <math>f_8 = - \sum_{i=2}^{2N_g}p_{x+y}(i)log(p_{x+y}(i))</math>
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* 9) Entropy: <math>f_9 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p(i,j))</math>
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* 10) Difference Variance: <math>f_{10} = \sum_{i=0}^{N_g-1} i^2 p_{x-y}(i)</math>
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* 11) Difference Entropy: <math>f_{11}= -\sum_{i=0}^{N_g-1} p_{x-y}(i)log(p_{x-y}(i))</math>
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* 12) Information Measures of Correlation 1: <math>f_{12} = \cfrac{HXY-HXY1}{max(HX,HY)}</math>
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* 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>,
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<math>HXY1 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p_x(i) p_y(i))</math>,
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<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>
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For more information about the Textural features, please refer to [1].
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== Reference ==
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* [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.

Latest revision as of 03:22, 27 April 2009

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

Notation

  --p(i,j): (i,j)th entry in a normalized gray-tone spatial dependence matrix, p(i,j) = P(i,j) / R
    * P(i,j) is the co-occurrence matrix and R is the sum of values in it, thus P(i,j) 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.

  --px(i) / py(i): ith entry in the marginal-probability distribution matrix obtained by summing the rows/columns of p(i,j).

  --Ng: Number of distinct gray levels in the image.



  --px + y(k):  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 
    px + y(i) is the probability of co-occurrence matrix coordinates summing to x+y

  --px − y(k):  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


Textural Features

  • 1) Angular Second Moment:f_1 = \sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)^2
  • 2) Contrast: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)}
  • 3) Correlation:f_3 = \cfrac{\sum_i \sum_j (i,j)p(i,j)-u_x u_y}{\sigma_x \sigma_y}, where ux,uy,σx,σy are the means and std.deviations of px and py, the partial probability density functions
  • 4) Sum of the Squares of Variance: f_4 = \sum_{i=1}^{N_g} \sum_{j=1}^{N_g} (i-j)^2 p(i,j)
  • 5) Inverse Difference Moment: f_5 = \sum_i \sum_j \cfrac{1}{1+(i-j)^2}p(i,j)
  • 6) Sum Average: f_6 = \sum_{i=2}^{2N_g} ip_{x+y}(i)
  • 7) Sum Variance: f_7 = \sum_{i=2}^{2N_g}(i-f_8)^2 p_{x+y}(i)
  • 8) Sum Entropy: f_8 = - \sum_{i=2}^{2N_g}p_{x+y}(i)log(p_{x+y}(i))
  • 9) Entropy: f_9 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p(i,j))
  • 10) Difference Variance: f_{10} = \sum_{i=0}^{N_g-1} i^2 p_{x-y}(i)
  • 11) Difference Entropy: f_{11}= -\sum_{i=0}^{N_g-1} p_{x-y}(i)log(p_{x-y}(i))
  • 12) Information Measures of Correlation 1: f_{12} = \cfrac{HXY-HXY1}{max(HX,HY)}
  • 13) Information Measures of Correlation 2: f13 = (1 − exp( − 2.0 | HXY2 − HXY | ))1 / 2, with HXY = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p(i,j)),

HXY1 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j)log(p_x(i) p_y(i)), HXY2 = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p_x(i)p_y(j)log(p_x(i) p_y(i))

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.
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