  We present a mathematical and algorithmic scheme for learning the principal
geometric elements in an image or 3D object. We build on recent work that
convexifies the basic problem of finding a combination of a small number shapes
that overlap and occlude one another in such a way that they "match" a given
scene as closely as possible. This paper derives general sufficient conditions
under which this convex shape composition identifies a target composition. From
a computational standpoint, we present two different methods for solving the
associated optimization programs. The first method simply recasts the problem
as a linear program, while the second uses the alternating direction method of
multipliers with a series of easily computed proximal operators.
