学习多面体范数与凸正则化，提升图像重建效果。
Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers
- 用可学习字典的l1惩罚合成形式，或可训练算子的l∞惩罚分析形式。
- 在生物医学图像重建中优于传统压缩感知方法，且收敛性相同。
- 适合图像重建、去噪任务，尤其对有限数据场景有效。
本文研究从有限数据中重建图像时学习凸正则化的方法。通过要求重建过程保持振幅等变性，将可接受的泛函类缩小为某个半范数的幂形式。进一步证明，这类泛函可由多面体范数任意逼近。文中提出两种对偶参数化：(i) 含可学习字典的合成形式，采用ℓ₁惩罚；(ii) 含可训练正则化算子的分析形式，采用ℓ∞惩罚。提供几何解释并证明两者具有通用性。提出基于紧框架与加权ℓ₁惩罚的具体实现架构，易于训练。在去噪和生物医学图像重建任务中验证，所提框架优于基于稀疏性的压缩感知方法，同时保持相同的收敛性和鲁棒性保证。
原文摘要 · Abstract (English)
This paper addresses the task of learning convex regularizers to guide the reconstruction of images from limited data. By imposing that the reconstruction be amplitude-equivariant, we narrow down the class of admissible functionals to those that can be expressed as a power of a seminorm. We then show that such functionals can be approximated to arbitrary precision with the help of polyhedral norms. In particular, we identify two dual parameterizations of such systems: (i) a synthesis form with an $\ell_1$-penalty that involves some learnable dictionary; and (ii) an analysis form with an $\ell_\infty$-penalty that involves a trainable regularization operator. After having provided geometric insights and proved that the two forms are universal, we propose an implementation that relies on a specific architecture (tight frame with a weighted $\ell_1$ penalty) that is easy to train. We illustrate its use for denoising and the reconstruction of biomedical images. We find that the proposed framework outperforms the sparsity-based methods of compressed sensing, while it offers essentially the same convergence and robustness guarantees.
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