Parseval Convolution Operators and Neural Networks

• URL: http://arxiv.org/abs/2408.09981v1
• Date: Mon, 19 Aug 2024 13:31:16 GMT
• Title: Parseval Convolution Operators and Neural Networks
• Authors: Michael Unser, Stanislas Ducotterd,
• Abstract summary: We first identify 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. We demonstrate the usage of those tools with the design of a CNN-based algorithm for the iterative reconstruction of biomedical images.
• Score: 16.78532039510369
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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