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Gridless 2D Recovery of Lines using the Sliding Frank-Wolfe Algorithm

Kévin Polisano, Basile Dubois-Bonnaire, Sylvain Meignen

TL;DR

This work tackles robust 2D line recovery from degraded measurements by casting line super-resolution as a sparse inverse problem over measures and solving it with Sliding Frank–Wolfe within the Beurling LASSO framework. It introduces two kernel models—Gaussian Line for blur deconvolution and Chirp Line for ridges in spectrograms—mapping line parameters to continuous atoms in a 2D search space. The proposed method achieves high-precision line parameter recovery with faster convergence than prior two-step approaches, avoiding discretization and enabling direct estimation in parameter space. The results demonstrate strong resilience to blur, noise, and interference, with potential extensions to denoising-first stages and patch-based reconstruction for more complex signals.

Abstract

We present a new approach leveraging the Sliding Frank--Wolfe algorithm to address the challenge of line recovery in degraded images. Building upon advances in conditional gradient methods for sparse inverse problems with differentiable measurement models, we propose two distinct models tailored for line detection tasks within the realm of blurred line deconvolution and ridge detection of linear chirps in spectrogram images.

Gridless 2D Recovery of Lines using the Sliding Frank-Wolfe Algorithm

TL;DR

This work tackles robust 2D line recovery from degraded measurements by casting line super-resolution as a sparse inverse problem over measures and solving it with Sliding Frank–Wolfe within the Beurling LASSO framework. It introduces two kernel models—Gaussian Line for blur deconvolution and Chirp Line for ridges in spectrograms—mapping line parameters to continuous atoms in a 2D search space. The proposed method achieves high-precision line parameter recovery with faster convergence than prior two-step approaches, avoiding discretization and enabling direct estimation in parameter space. The results demonstrate strong resilience to blur, noise, and interference, with potential extensions to denoising-first stages and patch-based reconstruction for more complex signals.

Abstract

We present a new approach leveraging the Sliding Frank--Wolfe algorithm to address the challenge of line recovery in degraded images. Building upon advances in conditional gradient methods for sparse inverse problems with differentiable measurement models, we propose two distinct models tailored for line detection tasks within the realm of blurred line deconvolution and ridge detection of linear chirps in spectrogram images.
Paper Structure (16 sections, 14 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 14 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. BLASSO and Frank--Wolfe Algorithms
  3. Beurling LASSO
  4. The Frank--Wolfe Algorithm
  5. Gaussian Lines Model
  6. Convolution of a Line with a Point Spread Function
  7. Gaussian Line Kernel
  8. Linear chirps as Lines in the Spectrogram
  9. Spectrogram of the superimposition of linear chirps
  10. Chirp Line Kernel
  11. Numerical experiments and Discussions
  12. Implementation details
  13. Gaussian Lines Estimation
  14. Chirp Lines Estimation
  15. Discussions
  16. ...and 1 more sections

Figures (3)

  • Figure 1: (a) Three Gaussian Lines with parameters of Experiment 1 (see Section \ref{['sec:GLest']}) in the noiseless case. (b) Its Radon transform in the plane $(\theta,\eta)$.
  • Figure 2: (a) Exp. 1 (very noisy lines), (b) Exp. 2 (very close lines plus noise), (c) Exp. 3 (more lines with different amplitudes plus noise). The estimated lines are depicted in red. (d) Ground truth spikes (in black) and estimated ones (in red) in the parameter space.
  • Figure 3: (a) Exp. 4 in the noiseless case, (b) Exp. 4 with no interference and high amount of noise ($\chi_2$ distributed), (c) Exp. 5 for crossing lines with interference and moderate noise, (d) Exp. 6 for parallel close lined with more interference and moderate noise. The estimated lines are depicted in red.

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4