Rearranging the operations

Rewriting Eq. 1 with the expansion of the filtered form and explicitly noting the iteration over average values for the $max$ operation, we obtain:

$$F_{i,j} = \bigvee_k \frac{1}{N_k} \sum_{m=-p}^{p} \sum_{n=-p}^{p} {E_k}_{m,n} * D_{i+m,j+n} ~~~\forall i,j$$ (3)

What we want is the filtered grid value at every point in $F$. So, $F$ is the set of all the filtered grid values:

$$F = \{ F_{i,j} \}$$ (4)

Let us now denote by ${F_k}_{i,j}$ the result of the linear operation:

$${F_k}_{i,j} = \sum_{m=-p}^{p} \sum_{n=-p}^{p} {E_k}_{m,n} * D_{i+m,j+n} ~~~\forall i,j$$ (5)

Recognizing that the ${F_k}_{i,j}$'s do not depend on each other, the max operation can be moved outside the braces, i.e. the maximum will be done after we have obtained all the pixels of the filtered image $F_k$. Then, the entire image $F$ can be written as:

$$F = \bigvee_{k,\forall i,j} \{ ( \frac{1}{N_k} * {F_k}_{i,j} ) \}$$ (6)

where $F_k$ is the image formed from the grid values, ${F_k}_{i,j}$.