Role of the rank of the filter mask matrix in image processing?
Separable just means you can do it in the x-direction and then in the y-direction and have it come out the same as if you did it in both dimensions simultaneously to begin with. It's not too hard to see that this will work for an average filter. If the filter is averaging over a 3x3 grid then in the 2-d case you take an average of nine values. In the separable filter case you first take three averages of three values. Then you average those three averages together. In both cases you get the same answer.

The separable case is much faster because you get to reuse some of the work you did in the x-dimension when you are doing the y-direction. In other words, each average of three values you computed in the x-direction will be used multiple times when filtering in the y-direction. The filters that can be made separable are precisely those whose matrix rank is one.