Variations on the CSC model

TL;DR

This paper broadens the convolutional sparse coding (CSC) framework by introducing two convex formulations that incorporate local signal characteristics, along with efficient algorithms for their solution.

Contribution

It proposes two new convex CSC formulations combining global and local penalties, and develops algorithms with proven convergence for these models.

Findings

New convex CSC formulations introduced

Algorithms with guaranteed convergence developed

Enhanced modeling of local signal features achieved

Abstract

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.