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Stable recovery guarantees for blind deconvolution under random mask assumption

Song Li, Yu Xia

TL;DR

This work tackles robust blind deconvolution with generalized random coded masks by formulating observations as $\boldsymbol{y}_l = \boldsymbol{h} \circledast (\boldsymbol{d}_l \odot \boldsymbol{x}) + \boldsymbol{z}_l$ under $\ell_2$-bounded noise, and extends the mask model beyond Rademacher distributions. It develops a constrained least squares framework that achieves near-optimal reconstruction error $\|\boldsymbol{X}^{\#} - \widehat{\boldsymbol{h}}\boldsymbol{x}^T\|_F \lesssim \sqrt{n}\|\boldsymbol{Z}\|_F$ when $L$ scales polylogarithmically with $n$, and proves a matching lower bound to establish near-optimality. For sparse $\boldsymbol{x}$, the authors show that robust recovery of the blur kernel $\boldsymbol{h}$ requires only $L = O(\log n)$ measurements under favorable conditions, after which a LASSO-based recovery of $\boldsymbol{x}$ yields provable error guarantees; a corollary further reduces requirements when $\boldsymbol{h}$ has compact support and sufficient sparsity separation. The PALM algorithm with carefully constructed initializations further improves practical performance, including in two-dimensional imaging scenarios, demonstrating enhanced robustness and accuracy over existing methods. Overall, the results advance blind deconvolution with coded masks by enabling broader mask families, noise-robust guarantees, and sample-efficient sparse-recovery strategies with concrete algorithmic realizations.

Abstract

This study addresses the blind deconvolution problem with modulated inputs, focusing on a measurement model where an unknown blurring kernel $\boldsymbol{h}$ is convolved with multiple random modulations $\{\boldsymbol{d}_l\}_{l=1}^{L}$(coded masks) of a signal $\boldsymbol{x}$, subject to $\ell_2$-bounded noise. We introduce a more generalized framework for coded masks, enhancing the versatility of our approach. Our work begins within a constrained least squares framework, where we establish a robust recovery bound for both $\boldsymbol{h}$ and $\boldsymbol{x}$, demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on $\boldsymbol{x}$. This approach significantly reduces the sampling complexity to the order of $L=O(\log n)$ when the non-zero elements of $\boldsymbol{x}$ are sufficiently separated. Furthermore, we demonstrate that incorporating sparsity constraints yields a refined error bound compared to the traditional constrained least squares model. The proposed method results in more robust and precise signal recovery, as evidenced by both theoretical analysis and numerical simulations. These findings contribute to advancing the field of blind deconvolution and offer potential improvements in various applications requiring signal reconstruction from modulated inputs.

Stable recovery guarantees for blind deconvolution under random mask assumption

TL;DR

This work tackles robust blind deconvolution with generalized random coded masks by formulating observations as under -bounded noise, and extends the mask model beyond Rademacher distributions. It develops a constrained least squares framework that achieves near-optimal reconstruction error when scales polylogarithmically with , and proves a matching lower bound to establish near-optimality. For sparse , the authors show that robust recovery of the blur kernel requires only measurements under favorable conditions, after which a LASSO-based recovery of yields provable error guarantees; a corollary further reduces requirements when has compact support and sufficient sparsity separation. The PALM algorithm with carefully constructed initializations further improves practical performance, including in two-dimensional imaging scenarios, demonstrating enhanced robustness and accuracy over existing methods. Overall, the results advance blind deconvolution with coded masks by enabling broader mask families, noise-robust guarantees, and sample-efficient sparse-recovery strategies with concrete algorithmic realizations.

Abstract

This study addresses the blind deconvolution problem with modulated inputs, focusing on a measurement model where an unknown blurring kernel is convolved with multiple random modulations (coded masks) of a signal , subject to -bounded noise. We introduce a more generalized framework for coded masks, enhancing the versatility of our approach. Our work begins within a constrained least squares framework, where we establish a robust recovery bound for both and , demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on . This approach significantly reduces the sampling complexity to the order of when the non-zero elements of are sufficiently separated. Furthermore, we demonstrate that incorporating sparsity constraints yields a refined error bound compared to the traditional constrained least squares model. The proposed method results in more robust and precise signal recovery, as evidenced by both theoretical analysis and numerical simulations. These findings contribute to advancing the field of blind deconvolution and offer potential improvements in various applications requiring signal reconstruction from modulated inputs.

Paper Structure

This paper contains 23 sections, 13 theorems, 63 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Related Works
  4. General Signal Case
  5. Subspace Signal or Sparse Signal Case.
  6. Our Contributions
  7. Notations and Definitions
  8. Orgainizations
  9. Main Results
  10. General Signal Case.
  11. Sparse Signal Case
  12. Refined Algorithm
  13. Numerical Experiments
  14. Experimental Setup
  15. General Signal Case
  16. ...and 8 more sections

Key Result

Theorem 2.1

Consider a sequence of independent diagonal matrices $\boldsymbol{D}_l = \mathrm{diag}({\boldsymbol d}_l)$, $l = 1, \dots, L$, where the diagonal entries are independent copies of a random variable $g \in \mathbb{C}$ with parameter $\nu$ as defined in Definition def: g. For any fixed $\widehat{\bold provided that $L \gtrsim C_{\nu} \mu \log^2 n \log\left({n}/{\mu}\right) \log\log n$ and $n \geq \m

Figures (5)

  • Figure 1: Average SNR of Reconstruction vs. Noise Level: (A) Real-Valued Case; (B) Complex-Valued Case
  • Figure 2: Performance Comparison of Different Models in the Sparse Scenario: (A) Real-Valued Case; (B) Complex-Valued Case.
  • Figure 3: Average SNR of Reconstruction vs. Noise Level in the Sparse Scenario: (A) Real-Valued Case; (B) Complex-Valued Case.
  • Figure 4: Original Image and Gaussian Filter: (a) Original Image; (b) Gaussian Filter.
  • Figure 5: Performance Comparison of Different Models in the Sparse Scenario: (a) Recovered Image by PALM algorithm, RMSE = 82.69 dB; (b) Recovered Image by Least Squares Method, RMSE = 15.27 dB.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.2
  • Remark 2.4
  • Theorem 2.3
  • Remark 2.5
  • Remark 2.6
  • ...and 12 more