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Five ways to recover the symbol of a non-binary localization operator

Simon Halvdansson

TL;DR

The paper tackles the inverse problem of recovering a non-binary symbol $f$ for localization operators $A_f^g$ within time-frequency analysis, extending beyond the classical binary-mask setting. It develops five constructive recovery strategies: recover via white noise, recover via spectral data (weighted accumulated Cohen's class and Wigner distributions), recover via plane tiling, and recover via Gabor projection; each comes with rigorous $L^1$-error bounds tied to the symbol’s variation and the operator’s spectral data. The results are framed in the quantum-harmonic-analysis formalism, yielding both theoretical guarantees and practical algorithms, along with numerical implementations and comparisons using MATLAB/LTFAT. Collectively, these methods enable robust, scalable symbol identification for non-binary masks, with potential benefits for calibration, artifact reduction, and flexible time-frequency design. The work advances the understanding of how operator-symbol mappings extend to non-binary regimes and provides concrete tools for analyzing and reconstructing time-frequency localization structures.

Abstract

Five constructive methods for recovering the symbol of a time-frequency localization operator with non-binary symbol are presented, two based on earlier work and three novel methods. For the two derivative methods which have previously been applied to binary symbols, we propose a changed symbol estimator and provide additional estimates that show how we can recover non-binary symbols. The three novel methods each have their own advantages and are all applicable to non-binary symbols. Two of them rely on prescribing the input of the localization operator and examining the output, allowing for targeting of the part of the symbol one wishes to recover while the last one relies on spectral information about the operator. All five methods are also implemented numerically and evaluated with the code available.

Five ways to recover the symbol of a non-binary localization operator

TL;DR

The paper tackles the inverse problem of recovering a non-binary symbol for localization operators within time-frequency analysis, extending beyond the classical binary-mask setting. It develops five constructive recovery strategies: recover via white noise, recover via spectral data (weighted accumulated Cohen's class and Wigner distributions), recover via plane tiling, and recover via Gabor projection; each comes with rigorous -error bounds tied to the symbol’s variation and the operator’s spectral data. The results are framed in the quantum-harmonic-analysis formalism, yielding both theoretical guarantees and practical algorithms, along with numerical implementations and comparisons using MATLAB/LTFAT. Collectively, these methods enable robust, scalable symbol identification for non-binary masks, with potential benefits for calibration, artifact reduction, and flexible time-frequency design. The work advances the understanding of how operator-symbol mappings extend to non-binary regimes and provides concrete tools for analyzing and reconstructing time-frequency localization structures.

Abstract

Five constructive methods for recovering the symbol of a time-frequency localization operator with non-binary symbol are presented, two based on earlier work and three novel methods. For the two derivative methods which have previously been applied to binary symbols, we propose a changed symbol estimator and provide additional estimates that show how we can recover non-binary symbols. The three novel methods each have their own advantages and are all applicable to non-binary symbols. Two of them rely on prescribing the input of the localization operator and examining the output, allowing for targeting of the part of the symbol one wishes to recover while the last one relies on spectral information about the operator. All five methods are also implemented numerically and evaluated with the code available.
Paper Structure (31 sections, 15 theorems, 79 equations, 9 figures, 1 table)

This paper contains 31 sections, 15 theorems, 79 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Time-frequency analysis
  4. Short-time Fourier transform
  5. Localization operators
  6. Gabor spaces
  7. Modulation spaces
  8. Cohen's class of time-frequency distributions
  9. Quantum harmonic analysis
  10. Mixed-state localization operators
  11. Fourier-Wigner transform
  12. Weyl quantization
  13. Cohen's class as operator-operator convolutions
  14. Functional analytic and probabilistic aspects of white noise
  15. Approximate identities
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $f \in C^{d+2}_c(\mathbb{R}^{2d})$ be real-valued, $\rho$ be given by eq:rho_def with white noise variance $\sigma^2$, $g, \varphi \in \mathcal{S}(\mathbb{R}^d)$ with $\Vert g \Vert_{L^2} = \Vert \varphi \Vert_{L^2} = 1$ and define where Then there exists a $C > 0$ such that where $B = AB_1 + B_2$ and $A$ is a constant independent of $f$ and $g$.

Figures (9)

  • Figure 1: Overview of all methods for a collection of four symbols.
  • Figure 2: All the intermediate steps in the white noise approximation detailed in Section \ref{['sec:white_noise_recovery']} for $\varphi = g$. The subscript on $\rho$ indicates the number of samples $K$ used.
  • Figure 3: A symbol, the associated accumulated spectrogram and the deconvolved estimate of the symbol.
  • Figure 4: Example of a deconvolution with visible errors.
  • Figure 5: Example of a failed deconvolution due to non-orthogonal eigenfunctions.
  • ...and 4 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 2.5: Romero2022
  • ...and 19 more