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Localization of point scatterers via sparse optimization on measures

Giovanni S. Alberti, Romain Petit, Matteo Santacesaria

TL;DR

The paper tackles locating point-like inhomogeneities in a medium from far-field acoustic data by reframing the Foldy-Lax inverse problem as a sparse-measure recovery task. It couples a linearized Beurling LASSO step, using the Born-approximation forward operator, with a subsequent nonlinear local descent to refine amplitudes and locations, leveraging off-grid sparse recovery theory for guarantees. The authors provide quantitative bounds on the linearization error that depend on scatterer separation and strength, establish stable recovery results (and exact support results under stronger conditions) for randomly sampled Fourier-like measurements, and demonstrate the method with numerical experiments, including two- and multi-scatterer scenarios. A JAX-based open-source implementation is offered, and the work suggests practical tradeoffs between frequency choice, separation, and measurement count, with a path forward for more advanced models and lifting techniques.

Abstract

We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy-Lax model, the estimation of the scatterers' locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a "linearize and locally optimize" approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.

Localization of point scatterers via sparse optimization on measures

TL;DR

The paper tackles locating point-like inhomogeneities in a medium from far-field acoustic data by reframing the Foldy-Lax inverse problem as a sparse-measure recovery task. It couples a linearized Beurling LASSO step, using the Born-approximation forward operator, with a subsequent nonlinear local descent to refine amplitudes and locations, leveraging off-grid sparse recovery theory for guarantees. The authors provide quantitative bounds on the linearization error that depend on scatterer separation and strength, establish stable recovery results (and exact support results under stronger conditions) for randomly sampled Fourier-like measurements, and demonstrate the method with numerical experiments, including two- and multi-scatterer scenarios. A JAX-based open-source implementation is offered, and the work suggests practical tradeoffs between frequency choice, separation, and measurement count, with a path forward for more advanced models and lifting techniques.

Abstract

We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy-Lax model, the estimation of the scatterers' locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a "linearize and locally optimize" approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.
Paper Structure (52 sections, 10 theorems, 67 equations, 5 figures, 1 algorithm)

This paper contains 52 sections, 10 theorems, 67 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Derivation of the Foldy-Lax model as a vanishing size limit.
  4. Related works
  5. Reconstruction methods in inverse acoustic scattering.
  6. Reconstruction of small inhomogeneities.
  7. Born approximation.
  8. Inverse scattering in the nonlinear case.
  9. Contributions
  10. Structure of the paper.
  11. Notations
  12. General notations.
  13. Forward operators.
  14. Green functions.
  15. Foldy matrix.
  16. ...and 37 more sections

Key Result

Proposition 4.1

Let $a\in(\mathbb{C}^*)^2$ and $x\in\mathcal{X}^2$. Define $\alpha=\kappa^2 |G(x_1,x_2)|\sqrt{|a_1||a_2|}$. If $\alpha<1$, then we have

Figures (5)

  • Figure 1: Dependance of the linearization error on $\kappa$ and $\Delta$. The dashed line corresponds to the bound \ref{['two_bound']}, and the plain line to the empirical error.
  • Figure 2: True linearization error $\|\Phi^f_x a-\Phi^b_x a\|_2$ and naive upper bound $\|\Phi^f_x a\|_2+\|\Phi^b_x a\|_2$ for $\kappa=1$. On the left of the vertical black line we have $|\beta|=\alpha>1$, and on the right $|\beta|=\alpha<1$.
  • Figure 3: Linear and nonlinear estimates for two unknown scatterers with several separation distances $\Delta$, obtained with $m=20$ measurements.
  • Figure 4: Linear and nonlinear estimates for nine unknown scatterers with $m=100$ measurements and Gaussian measurement noise with standard deviation $0.1$.
  • Figure 5: Output of the nonlinear step initialized with a measure supported on a $4\times 4$ (top row) and a $5\times 5$ (bottom row) regular grid. The experimental setup is the same as in \ref{['fig:exp_2']}.

Theorems & Definitions (20)

  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • Theorem 4.4
  • Remark 4.5
  • Remark 4.6
  • Proposition A.1
  • proof
  • ...and 10 more