Maximum weight digital regions decomposable into digital star-shaped regions

We consider an optimization version of the image segmentation problem, in which we are given a grid graph with weights on the grid cells. We are interested in finding the maximum weight subgraph such that the subgraph can be decomposed into two "star-shaped" images. We show that this problem can be reduced to the problem of finding a maximum-weight closed set in an appropriately defined directed graph which is well known to have efficient algorithms which run very fast in practice. We also show that finding a maximum-weight subgraph that is decomposable into m star-shaped objects is NP-hard for some m 2.