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A multilevel framework for accelerating uSARA in radio-interferometric imaging

Guillaume Lauga, Audrey Repetti, Elisa Riccietti, Nelly Pustelnik, Paulo Gonçalves, Yves Wiaux

Abstract

This paper presents a multilevel algorithm specifically designed for radio-interferometric imaging in astronomy. The proposed algorithm is used to solve the uSARA (unconstrained Sparsity Averaging Reweighting Analysis) formulation of this image restoration problem. Multilevel algorithms rely on a hierarchy of approximations of the objective function to accelerate its optimization. In contrast to the usual multilevel approaches where this hierarchy is derived in the parameter space, here we construct the hierarchy of approximations in the observation space. The proposed approach is compared to a reweighted forward-backward procedure, which is the backbone iteration scheme for solving the uSARA problem.

A multilevel framework for accelerating uSARA in radio-interferometric imaging

Abstract

This paper presents a multilevel algorithm specifically designed for radio-interferometric imaging in astronomy. The proposed algorithm is used to solve the uSARA (unconstrained Sparsity Averaging Reweighting Analysis) formulation of this image restoration problem. Multilevel algorithms rely on a hierarchy of approximations of the objective function to accelerate its optimization. In contrast to the usual multilevel approaches where this hierarchy is derived in the parameter space, here we construct the hierarchy of approximations in the observation space. The proposed approach is compared to a reweighted forward-backward procedure, which is the backbone iteration scheme for solving the uSARA problem.
Paper Structure (12 sections, 1 theorem, 10 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 1 theorem, 10 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The multilevel framework
  3. Proposed coarse model in data space
  4. ML approach for uSARA acceleration
  5. uSARA approach in a nutshell
  6. Proposed IML-FISTA for uSARA
  7. Algorithmic settings and implementation
  8. Numerical experiments
  9. Simulated data
  10. Minimization comparison without reweighting
  11. Minimization comparison for uSARA
  12. Conclusion and perspectives

Key Result

Theorem 1

Let $i\in \{1, \ldots, I\}$. Let $(x_k)_{k\in \mathbb N}$ and $(z_k)_{k\in \mathbb N}$ be sequences generated by algorithm algo:full-rw. Assume that, for every $k\in \mathbb N$, the coarse model defined in Algorithm alg:IMLFISTA decreases, i.e. $F_H(z_+) \leq F_H(z_k)$This is ensured as soon as $\ta

Figures (3)

  • Figure 1: $u-v$ coverage for the fine (in blue) and coarse level (in red) when using the MeerKAT telescope MeerKAT.
  • Figure 2: Evolution of the objective function values (left), of the SNR (middle) of the iterates produced by FB algorithm, FISTA and IML-FISTA with respect to the CPU time when solving Problem \ref{['min:fine-gen']} (each algorithm had a CPU time budget of 1500 seconds). Evolution of the SNR for the three algorithms when we involve the complete reweighting procedure (right) (CPU time budget of 10000 seconds).
  • Figure 3: Reconstruction in log scale of a region of the M31 galaxy by FB (top row) FISTA (middle row) and IML-FISTA (bottom row) at equivalent CPU times. The legend on top of each thumbnail reads as follows: log SNR in dB - CPU time in seconds. log SNR = SNR($\log_{10}(10^3 x + 1)/3, \log_{10}(10^3 x_{\text{truth}} + 1)/3$).

Theorems & Definitions (1)

  • Theorem 1