  Over the past decade, the celebrated sparse representation model has achieved
impressive results in various signal and image processing tasks. A
convolutional version of this model, termed convolutional sparse coding (CSC),
has been recently reintroduced and extensively studied. CSC brings a natural
remedy to the limitation of typical sparse enforcing approaches of handling
global and high-dimensional signals by local, patch-based, processing. While
the classic field of sparse representations has been able to cater for the
diverse challenges of different signal processing tasks by considering a wide
range of problem formulations, almost all available algorithms that deploy the
CSC model consider the same $\ell_1 - \ell_2$ problem form. As we argue in this
paper, this CSC pursuit formulation is also too restrictive as it fails to
explicitly exploit some local characteristics of the signal. This work expands
the range of formulations for the CSC model by proposing two convex
alternatives that merge global norms with local penalties and constraints. The
main contribution of this work is the derivation of efficient and provably
converging algorithms to solve these new sparse coding formulations.
