Local simplicity, topology, and sets of finite perimeter

Abstract

We introduce a useful and relatively easy to check condition, local simplicity, which provides significantly more structure to sets of finite perimeter, while not being too restrictive. Local simplicity holds for minimizers in a wide variety of variational problems in materials science, biology, image processing, oncology, and other fields. We prove several regularity and structural properties of locally simple sets and their boundaries, including a vital decomposition theorem that in our setting strengthens the conclusion of theorems of H. Federer and L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel. We establish strong connections between topology and sets of finite perimeter, so that ordinary notions of openness, closedness, and connectedness may be readily used in the finite perimeter setting.
We apply these results to an image reconstruction procedure from image processing, _L_1 TV-minimization. The density ratio bounds computed to establish local simplicity are themselves of practical importance, as they provide concrete, easy to compute criteria to check simulations against.