How to apply a blur only to a rectangular area within a pane?

Let's say I've got a StackPane which has a BackgroundImage as background and another StackPane (or another component, if neccessary) as a child. The child covers only a part of the parent StackPane.

I'd like to know how to apply a GaussianBlur just to the area the child covers, so that the BackgroundImageis blurry in this area.

The size of the child changes when the parent is resized. It would be perfect to get a solution that will resize just in time, too.