Decomposing a shape into visually meaningful parts comes naturally to humans, but recreating this fundamental operation
in computers has been shown to be difficult. Similar challenges have puzzled researchers in shape reconstruction
for decades. In this paper, we recognize the strong connection between shape reconstruction and shape decomposition
at a fundamental level and propose a method called alpha-decomposition. The alpha-decomposition generates a space of decompositions
parametrized by alpha, the diameter of a circle convolved with the input polygon. As we vary the value of alpha,
some structural features appear and disappear quickly while others persist. Therefore, by analyzing the persistence of
the features, we can determine better decompositions that are more robust to both geometrical and topological noise.