  The traditional sparse modeling approach, when applied to inverse problems
with large data such as images, essentially assumes a sparse model for small
overlapping data patches. While producing state-of-the-art results, this
methodology is suboptimal, as it does not attempt to model the entire global
signal in any meaningful way - a nontrivial task by itself. In this paper we
propose a way to bridge this theoretical gap by constructing a global model
from the bottom up. Given local sparsity assumptions in a dictionary, we show
that the global signal representation must satisfy a constrained
underdetermined system of linear equations, which can be solved efficiently by
modern optimization methods such as Alternating Direction Method of Multipliers
(ADMM). We investigate conditions for unique and stable recovery, and provide
numerical evidence corroborating the theory.
