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.

中文翻译：

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卷积稀疏编码模型的变化

在过去十年中，著名的稀疏表示模型在各种信号和图像处理任务中取得了令人瞩目的成果。该模型的卷积版本，称为卷积稀疏编码 (CSC)，最近被重新引入并进行了广泛研究。CSC 为通过局部、基于补丁的处理处理全局和高维信号的典型稀疏执行方法的局限性带来了自然补救。虽然稀疏表示的经典领域已经能够通过考虑广泛的问题公式来满足不同信号处理任务的不同挑战，但几乎所有部署 CSC 模型的可用算法都考虑相同的 $\ell _1 - \ell _2 $ 问题表。正如我们在本文中论证的那样，这种 CSC 追踪公式也过于严格，因为它未能明确利用信号的某些局部特征。这项工作通过提出两个将全局规范与局部惩罚和约束合并的凸替代方案，扩展了 CSC 模型的公式范围。这项工作的主要贡献是推导出有效且可证明收敛的算法来解决这些新的稀疏编码公式。