We first establish a kernel theorem that characterizes all linear shift-invariant (LSI) operators acting on discrete multicomponent signals. This result naturally leads to the identification of the Parseval convolution operators as the class of energy-preserving filterbanks. We then present a constructive approach for the design/specification of such filterbanks via the chaining of elementary Parseval modules, each of which being parameterized by an orthogonal matrix or a 1-tight frame. Our analysis is complemented with explicit formulas for the Lipschitz constant of all the components of a convolutional neural network (CNN), which gives us a handle on their stability. Finally, we demonstrate the usage of those tools with the design of a CNN-based algorithm for the iterative reconstruction of biomedical images. Our algorithm falls within the plug-and-play framework for the resolution of inverse problems. It yields better-quality results than the sparsity-based methods used in compressed sensing, while offering essentially the same convergence and robustness guarantees.

翻译：我们首先建立了一个核定理，该定理刻画了作用于离散多分量信号的所有线性平移不变（LSI）算子。这一结果自然地将Parseval卷积算子识别为能量保持滤波器组这一类别。随后，我们提出了一种通过级联基本Parseval模块来设计/指定此类滤波器组的构造性方法，其中每个模块由一个正交矩阵或一个1-紧框架参数化。我们的分析辅以卷积神经网络（CNN）所有组成部分的Lipschitz常数的显式公式，这为我们把握其稳定性提供了依据。最后，我们通过设计一种基于CNN的生物医学图像迭代重建算法，展示了这些工具的应用。我们的算法属于用于求解逆问题的即插即用框架。与压缩感知中使用的基于稀疏性的方法相比，该算法能产生质量更优的结果，同时提供基本相同的收敛性和鲁棒性保证。