\sum_{i,j}j \cdot g(i, j)$ (due to matrix symmetry), and $\g_{x+y}(k)=
\sum_{i,j}j \cdot g(i, j)$ (due to matrix symmetry).
\sum_{i}\sum_{j}g(i)$ where $ i+j=k $ and $ k =2, 3, .., 2N_[g}$ and $\g_{x-y}(k)=\sum_{i}\sum_{j}g(i)$ where $ i-j=k $ and $ k =0, 1, ..,N_[g}-1$
$N_{g}$ : Number of distinct gray levels in the quantized image.
$ g_{x+y}(k)=\sum_{i}\sum_{j}g(i)$ where $ i+j=k $ and $ k =2, 3, .., 2N_{g}$ and $ g_{x-y}(k)=\sum_{i}\sum_{j}g(i)$ where $ i-j=k $ and $ k =0, 1, ..,N_{g}-1$