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Parseval Convolution Operators and Neural Networks

Michael Unser, Stanislas Ducotterd

TL;DR

This work targets stability and reliability in CNN-based inverse problems by focusing on Parseval convolution operators for multichannel signals. It develops a kernel theorem that characterizes linear shift-invariant vector-valued operators and introduces a constructive, modular design of Parseval filterbanks via chaining unitary or 1-tight-frame modules. The authors quantify stability through explicit Lipschitz constants and demonstrate plug-and-play reconstructions for biomedical imaging, achieving higher-quality results than sparsity-based methods while preserving convergence guarantees. The framework enables the design of ultra-stable, energy-preserving CNNs that maintain robustness under composition, making them well-suited for trustworthy image reconstruction and related inverse problems.

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.

Parseval Convolution Operators and Neural Networks

TL;DR

This work targets stability and reliability in CNN-based inverse problems by focusing on Parseval convolution operators for multichannel signals. It develops a kernel theorem that characterizes linear shift-invariant vector-valued operators and introduces a constructive, modular design of Parseval filterbanks via chaining unitary or 1-tight-frame modules. The authors quantify stability through explicit Lipschitz constants and demonstrate plug-and-play reconstructions for biomedical imaging, achieving higher-quality results than sparsity-based methods while preserving convergence guarantees. The framework enables the design of ultra-stable, energy-preserving CNNs that maintain robustness under composition, making them well-suited for trustworthy image reconstruction and related inverse problems.

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.
Paper Structure (27 sections, 10 theorems, 84 equations, 2 figures, 3 tables)

This paper contains 27 sections, 10 theorems, 84 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works and Concepts
  3. Road Map
  4. Mathematical Background
  5. Notation
  6. The Discrete Fourier Transform and Plancherel's Isomorphism
  7. 1-Lip and Parseval Operators
  8. Vector-Valued LSI Operators on $\ell^N_2( \mathbb{Z}^d)$
  9. Refresher: Scalar Convolution Operators
  10. Multichannel Convolution Operators
  11. Kernel Theorem for Multichannel LSI Operators
  12. Parseval Filterbanks
  13. Parametrization of Parseval Filterbanks
  14. Normalized patch operator
  15. Parametric $1$-to-$N$ Parseval Module
  16. ...and 12 more sections

Key Result

proposition 1

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be two Hilbert spaces. Then, the linear operator ${\mathrm{T}}: {\mathcal{X}} \to {\mathcal{Y}}$ is a Parseval operator if any of the following equivalent conditions holds.

Figures (2)

  • Figure 1: Ground truth, zero-fill reconstruction $\vec{H}^{\mathsf{T}} \vec{y}$, and PnP-FBS reconstruction using several network parameterizations on the Brain image with the Cartesian mask. Lower panel: zoom of a region of interest. The SNR is evaluated with respect to the groundtruth (left image) and is overlaid in white.
  • Figure 2: Ground truth, zero-fill reconstruction $\vec{H}^{\mathsf{T}} \vec{y}$, and PnP-FBS reconstruction using several network parameterizations on the Bust image with the Cartesian mask. Lower panel: zoom of a region of interest. The SNR is evaluated with respect to the groundtruth (left image) and is overlaid in white.

Theorems & Definitions (21)

  • definition 1
  • definition 2
  • definition 3
  • proposition 1: Properties of Parseval operators
  • proof
  • proposition 2
  • definition 4
  • theorem 1: Kernel theorem for discrete LSI operators on $\ell_2( \mathbb{Z}^d)$
  • proof
  • theorem 2: Kernel theorem for LSI operators $\ell^N_2( \mathbb{Z}^d)\to \ell^M_2( \mathbb{Z}^d)$
  • ...and 11 more