  The convolutional sparse model has recently gained increasing attention in
the signal and image processing communities, and several methods have been
proposed for solving the pursuit problem emerging from it -- in particular its
convex relaxation, Basis Pursuit. In the first of this two-part work, we have
provided a theoretical back-bone for this model, providing guarantees for the
uniqueness of the sparsest solution and for the success of pursuit algorithms
by introducing the notion of stripe sparsity and other related measures.
Herein, we extend the analysis to a noisy regime, thereby considering signal
perturbations and model deviations. We address questions of stability of the
sparsest solutions and the success of pursuit algorithms, both greedy and
convex. Classical definitions such as the RIP are generalized to the
convolutional model, and existing notions such as the ERC are connected to our
setting. On the algorithmic side, we demonstrate how to solve the global
pursuit problem by using simple local processing, thus offering a first of its
kind bridge between global modeling of signals and their patch-based local
treatment.
