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Signal and Image Recovery with Scale and Signed Permutation Invariant Sparsity-Promoting Functions

Jianqing Jia, Ashley Prater-Bennette, Lixin Shen

TL;DR

This work advances sparse recovery and image restoration by leveraging scale- and signed permutation-invariant sparsity-promoting functions $h_p(\boldsymbol{x})$ with $p\in\{1,2\}$ and their proximal operators. By integrating these proximals into APG, FBS, and ADMM, the authors provide convergence analyses and demonstrate improved recovery performance under high coherence and dynamic range, as well as faster convergence. Key contributions include closed-form proximal for $p=2$, an efficient ADMM scheme (H2-ADMM) with Sherman–Morrison–Woodbury acceleration, and extensive numerical experiments showing superior accuracy and computational efficiency in both sparse signal recovery and TV-based image restoration. The results underscore the practical impact of these proximal-operator-enabled nonconvex formulations in challenging sensing and imaging tasks, with potential extensions to broader medical imaging and learning applications.

Abstract

Sparse signal recovery has been a cornerstone of advancements in data processing and imaging. Recently, the squared ratio of $\ell_1$ to $\ell_2$ norms, $(\ell_1/\ell_2)^2$, has been introduced as a sparsity-prompting function, showing superior performance compared to traditional $\ell_1$ minimization, particularly in challenging scenarios with high coherence and dynamic range. This paper explores the integration of the proximity operator of $(\ell_1/\ell_2)^2$ and $\ell_1/\ell_2$ into efficient optimization frameworks, including the Accelerated Proximal Gradient (APG) and Alternating Direction Method of Multipliers (ADMM). We rigorously analyze the convergence properties of these algorithms and demonstrate their effectiveness in compressed sensing and image restoration applications. Numerical experiments highlight the advantages of our proposed methods in terms of recovery accuracy and computational efficiency, particularly under noise and high-coherence conditions.

Signal and Image Recovery with Scale and Signed Permutation Invariant Sparsity-Promoting Functions

TL;DR

This work advances sparse recovery and image restoration by leveraging scale- and signed permutation-invariant sparsity-promoting functions with and their proximal operators. By integrating these proximals into APG, FBS, and ADMM, the authors provide convergence analyses and demonstrate improved recovery performance under high coherence and dynamic range, as well as faster convergence. Key contributions include closed-form proximal for , an efficient ADMM scheme (H2-ADMM) with Sherman–Morrison–Woodbury acceleration, and extensive numerical experiments showing superior accuracy and computational efficiency in both sparse signal recovery and TV-based image restoration. The results underscore the practical impact of these proximal-operator-enabled nonconvex formulations in challenging sensing and imaging tasks, with potential extensions to broader medical imaging and learning applications.

Abstract

Sparse signal recovery has been a cornerstone of advancements in data processing and imaging. Recently, the squared ratio of to norms, , has been introduced as a sparsity-prompting function, showing superior performance compared to traditional minimization, particularly in challenging scenarios with high coherence and dynamic range. This paper explores the integration of the proximity operator of and into efficient optimization frameworks, including the Accelerated Proximal Gradient (APG) and Alternating Direction Method of Multipliers (ADMM). We rigorously analyze the convergence properties of these algorithms and demonstrate their effectiveness in compressed sensing and image restoration applications. Numerical experiments highlight the advantages of our proposed methods in terms of recovery accuracy and computational efficiency, particularly under noise and high-coherence conditions.

Paper Structure

This paper contains 15 sections, 3 theorems, 58 equations, 6 figures, 5 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Application: Sparse Signal Recovery
  4. Algorithms: FBS and APG
  5. Algorithm: Alternating Direction Method of Multipliers (ADMM)
  6. Numerical Experiments: Sparse Signal Recovery
  7. Algorithmic Comparison
  8. Noise-free Measurements
  9. Noisy Data
  10. Application: Image Restoration
  11. Algorithm: Image Restoration
  12. Numerical Experiments: Image Restoration
  13. Experimental Settings
  14. Qualitative and Quantitative Comparisons
  15. Conclusion

Key Result

Lemma 3.1

Let $f$ be a proper function with Lipschitz continuous gradients and $g$ be proper and lower semicontinuous. For nonconvex $f$ and nonconvex nonsmooth $g$, assume that $F(\bm{x})=f(\bm{x})+ g(\bm{x})$ coercive. Then $\{\bm{x}^k\}$ and $\{\bm{v}^k\}$ generated by Algorithm alg:APG_CS are bounded. Let

Figures (6)

  • Figure 1: Algorithmic comparison of objective values for problem \ref{['model:uncons']} using DCT sensing matrices, presented in a $2 \times 2$ grid format. Rows correspond to $E=1,10$ from top to bottom, and columns correspond to $D=3,5$ from left to right.
  • Figure 2: Algorithmic comparison of relative error for problem \ref{['model:uncons']} with DCT sensing matrices, presented in a $2 \times 2$ grid format. Rows correspond to $E=1, 10$ from top to bottom, and columns correspond to $D=3, 5$ from left to right.
  • Figure 3: Algorithmic comparison of success rates is conducted for DCT sensing matrices. The presentation of results is structured in a $3 \times 3$ grid format, where rows correspond to $E=1,10, 20$ from top to bottom, and columns correspond to $D=0,1,5$ from left to right.
  • Figure 4: Original images: (a) Barbara, (b) Finger Print.
  • Figure 5: Quantitative comparisons of deblurring results: (a) degradation without noise, (b) restoration using the TV-PAPC method, and (c) restoration using the h2-ADMM method.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Lemma 3.1: Theorem 1 in Li-Lin:NISP:2015
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1