Meaning of textures Could someone explain me what does the "Sum Average" mean in equation 6th in the following link.
http://murphylab.web.cmu.edu/publications/boland/boland_node26.html
Its related to creation of image texture (2d array) from an image (2d array).
Thank you.
 A: This is horrible notation. It's defined in this paper. The symbol $p$ is multiply overloaded with various meanings that are inconsistent when taken literally according to standard notational conventions. The symbol $p_{x+y}(k)$ doesn't refer to a quantity indexed by $x+y$; it's only a function of $k$, with "$x+y$" merely indicating that this is the probability for the sum of the indices to take the value $k$; that is, in conventional notation, $p_{x+y}(k)=\textsf{Pr}(X+Y=k)$. In terms of the probabilities $p(i,j)$ in the co-occurence matrix, $p_{x+y}(k)=\sum_{i+j=k}p(i,j)$. Thus, the quantity referred to as the "sum average", $\sum_kkp_{x+y}(k)$, is the expected value of the sum of the gray levels of adjacent pixels.
A: From  the paper Joriki references, Appendix I, the authors define $4$ marginal-probability matrices (vectors) from:

$p(i,j)=(i,j)^{th}$-entry in a normalized gray-tone spatial-dependence matrix.

Namely:
$$p_x(i)=\sum\limits_{j=1}^{N_g}p(i,j)$$
$$p_y(j)=\sum\limits_{i=1}^{N_g}p(i,j)$$
$$p_{x+y}(k)=\sum\limits_{i=1}^{N_g}\sum\limits_{j=1, i+j=k}^{N_g}p(i,j)$$
$$p_{x-y}(k)=\sum\limits_{i=1}^{N_g}\sum\limits_{j=1, |i-j|=k}^{N_g}p(i,j)$$
which give sums across the horizontal, vertical and both diagonals.
So I am guessing the authors meant to define all $4$ 'sum averages' using the standard definition of expection, $E(X)=\sum xP(X=x)$.
