Algorithms for computing the maximum weight region decomposable into elementary shapes

Motivated by the image segmentation problem, we consider the following geometric optimization problem: Given a weighted n × n pixel grid, find the maximum weight region whose shape is decomposable into a set of disjoint elementary shapes. We give efficient algorithms for several interesting shapes. This is in strong contrast to finding the maximum weight region that is the union of elementary shapes for the corresponding cases—a problem that we prove to be NP-hard. We implemented one of the algorithms and demonstrate its applicability for image segmentation.