Extracting the Principal Shape Components via Convex Programming

We present a general method for extracting a region from an image (or 3D object) that can be expressed, or approximated, by taking unions and set differences from a collection of template shapes in a dictionary. We build on recent work that shows how this geometric problem can be recast in the language of linear algebra, with set operations on shapes translated into linear combinations of vectors, and solved using convex programming. This paper presents a set of sufficient conditions for which this convex program returns the "correct" shape. These conditions are robust in that they can account for the shapes that have indistinct boundaries, or model mismatch between the shapes in the dictionary and the target region in the image. We also present two different methods for solving the convex extraction program. 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. We present a number of numerical experiments that use the framework to perform image segmentation, optical character recognition, and find multi-resolution geometrical descriptions of 3D objects.