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GSW: Generalized "Self-Wiener" Denoising

Amir Weiss

Abstract

We revisit the recently proposed ``self-Wiener" (SW) filtering method for robust deconvolution, and generalize it to the classical denoising problem. The resulting estimator, termed generalized SW (GSW) filtering, retains the nonlinear shrinkage structure of SW but introduces a tunable threshold parameter. This tunability enables GSW to flexibly adapt to varying signal-to-noise ratio (SNR) regimes by balancing noise suppression and signal preservation. We derive closed-form expressions for its mean-square error (MSE) performance in both low- and high-SNR regimes, and demonstrate that GSW closely approximates the oracle MMSE at high SNR while maintaining strong robustness at low SNR. Simulation results validate the analytical findings, showing that GSW consistently achieves favorable denoising performance across a wide range of SNRs. Its analytical tractability, parameter flexibility, and close connection to the optimal Wiener filter structure make it a promising tool for practical applications including compressive sensing, sparse signal recovery, and domain-specific shrinkage in wavelet, Fourier, and potentially learned orthonormal representations.

GSW: Generalized "Self-Wiener" Denoising

Abstract

We revisit the recently proposed ``self-Wiener" (SW) filtering method for robust deconvolution, and generalize it to the classical denoising problem. The resulting estimator, termed generalized SW (GSW) filtering, retains the nonlinear shrinkage structure of SW but introduces a tunable threshold parameter. This tunability enables GSW to flexibly adapt to varying signal-to-noise ratio (SNR) regimes by balancing noise suppression and signal preservation. We derive closed-form expressions for its mean-square error (MSE) performance in both low- and high-SNR regimes, and demonstrate that GSW closely approximates the oracle MMSE at high SNR while maintaining strong robustness at low SNR. Simulation results validate the analytical findings, showing that GSW consistently achieves favorable denoising performance across a wide range of SNRs. Its analytical tractability, parameter flexibility, and close connection to the optimal Wiener filter structure make it a promising tool for practical applications including compressive sensing, sparse signal recovery, and domain-specific shrinkage in wavelet, Fourier, and potentially learned orthonormal representations.

Paper Structure

This paper contains 11 sections, 2 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Generalized "Self-Wiener" Filtering
  4. Definition and Intuition
  5. Asymptotic MSE Analysis
  6. High-SNR regime
  7. Low SNR regime
  8. Simulation Results
  9. Conclusion
  10. Proof of Proposition \ref{['thm:highSNR_GSW']}
  11. Proof of Proposition \ref{['thm:lowSNR_GSW']}

Key Result

Proposition 1

Let $z_n$ be as in eq:zn_def and define where $Q_1(\cdot,\cdot)$ is the Marcum $Q$-function corazza2002new. Then, for $|\eta_n| \gg \lambda$, the MSE of the estimator eq:GSWest admits the expansion

Figures (1)

  • Figure 1: MSE versus inverse noise RMS power for a sparse signal with $K = 10$ non-zero entries (unit magnitude) out of $N = 1000$. Results are averaged over $10^3$ realizations. GSW approaches the oracle MMSE bound at high SNR and outperforms LS, JS, ST, and SW at medium-to-high SNR.

Theorems & Definitions (4)

  • Remark 1: Relation to the original SW estimator
  • Proposition 1: High-SNR performance
  • Proposition 2: Low-SNR performance
  • Remark 2: Real-valued case