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arxiv: 2605.23784 · v1 · pith:QELCVMRAnew · submitted 2026-05-22 · 🧮 math-ph · math.MP

Reconstruction methods for inverse scattering problems with phaseless data

John C. Schotland , Shenwen Yu This is my paper

Pith reviewed 2026-05-25 02:42 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords inverse scatteringphaseless datainverse Born seriesSchrödinger equationfar-field scatteringphase retrievalscattering reciprocity
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The pith

Inverse Born series methods reconstruct scattering potentials from phaseless total and scattered field data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops reconstruction techniques for the inverse scattering problem governed by the Schrödinger equation when only the magnitude of the field is measured. It extends the inverse Born series to accommodate phaseless total-field measurements and determines the radius within which the series converges. A separate Fourier method uses reciprocity between incident and observation directions to extract coefficients of the scattering potential in the far field. For scattered-field data a polarization step restores the missing phase so that the standard inverse Born series can be applied. Numerical tests confirm that the resulting reconstructions match known potentials.

Core claim

We extend the IBS framework and analyze its radius of convergence for phaseless total-field data. In the far-field setting we propose a Fourier-based reconstruction method that exploits the scattering reciprocity between incident and observation directions to recover Fourier coefficients of the scattering potential. For phaseless scattered-field data we employ a polarization-based approach to recover phase information and enable IBS reconstruction.

What carries the argument

The inverse Born series (IBS) extended to phaseless total-field data, together with a reciprocity-based Fourier coefficient recovery step and a polarization-based phase restoration step for scattered-field data.

If this is right

  • Fourier coefficients of the scattering potential are recoverable from far-field reciprocity relations without phase information.
  • Polarization restores sufficient phase data to permit standard IBS reconstruction from scattered-field measurements.
  • The radius of convergence of the extended IBS is finite and can be characterized for each phaseless data type.
  • Numerical validation demonstrates that the reconstructed potentials agree with the true scatterers for the tested cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reciprocity and polarization steps could be combined with other iterative solvers beyond the Born series.
  • If the convergence radius is small, hybrid schemes that switch to nonlinear optimization after a few IBS terms may improve accuracy.
  • The methods assume exact reciprocity and polarization properties; small violations from noise or anisotropy would require separate stability analysis.

Load-bearing premise

The inverse Born series converges when applied directly to the three forms of phaseless data considered.

What would settle it

Numerical simulation of the extended IBS series on a known compactly supported potential that shows divergence or incorrect recovery once the data magnitude is taken.

Figures

Figures reproduced from arXiv: 2605.23784 by John C. Schotland, Shenwen Yu.

Figure 1
Figure 1. Figure 1: Reconstructions of low contrast disk using direct method 14 [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reconstructions of high contrast disk using direct method 15 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstructions of low contrast Gaussian mixture using direct method 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reconstructions of high contrast Gaussian mixture using direct method 17 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reconstructions of low contrast disk using Fourier method 18 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstructions of high contrast disk using Fourier method 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reconstructions of low contrast Gaussian mixture using Fourier method 20 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reconstructions of high contrast Gaussian mixture using Fourier method 21 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Reconstructions of low contrast disk using polarization method 23 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Reconstructions of high contrast disk using polarization method 24 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Reconstructions of low contrast Gaussian mixture using polar￾ization method 25 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Reconstructions of high contrast Gaussian mixture using polar￾ization method 26 [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
read the original abstract

We investigate phaseless inverse scattering problem for the Schr\"odinger equation and develop reconstruction methods based on the inverse Born series (IBS). We consider three types of phaseless data: the far-field total field, the total field and the far-field scattered field. For phaseless total-field data, we extend the IBS framework and analyze its radius of convergence. In the far-field setting, we propose a Fourier-based reconstruction method that exploits the scattering reciprocity between incident and observation directions to recover Fourier coefficients of the scattering potential. For phaseless scattered-field data, we employ a polarization-based approach to recover phase information and enable IBS reconstruction. Numerical experiments are conducted to validate the proposed methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript develops reconstruction methods for phaseless inverse scattering problems for the Schrödinger equation based on the inverse Born series (IBS). For phaseless total-field data, the IBS framework is extended and its radius of convergence is analyzed. In the far-field setting, a Fourier-based method exploits scattering reciprocity between incident and observation directions to recover Fourier coefficients of the scattering potential. For phaseless scattered-field data, a polarization-based approach recovers phase information to enable IBS reconstruction. Numerical experiments are presented to validate the proposed methods.

Significance. If the extensions, convergence analysis, and reconstruction techniques hold under the stated assumptions, the work would provide useful practical tools for inverse scattering where phase information is unavailable. The reciprocity-based Fourier recovery and polarization phase recovery represent targeted adaptations of existing IBS methods. Numerical validation is included, though the abstract provides no quantitative error bounds or convergence rates.

minor comments (1)
  1. [Abstract] The abstract claims numerical validation but supplies no information on the specific error metrics, noise models, domain sizes, or comparison methods employed in the experiments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for providing a concise summary of the work. We note that the report lists no specific major comments and gives an 'uncertain' recommendation. We would be pleased to address any concrete points the referee may wish to raise.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends the existing inverse Born series framework to three types of phaseless data, analyzes its radius of convergence, and introduces reconstruction techniques based on scattering reciprocity for Fourier recovery and polarization for phase recovery. These steps are presented as extensions and applications of the prior IBS method rather than reductions of the target results to fitted parameters or self-referential definitions. No load-bearing equations or claims in the abstract reduce by construction to the inputs; the numerical validation further indicates independent content. Self-citation of the IBS framework, if present, is not load-bearing for the new phaseless extensions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on standard domain assumptions in inverse scattering theory regarding convergence of the Born series and properties of the scattering potential; no explicit free parameters, invented entities, or ad-hoc axioms are stated.

axioms (1)
  • domain assumption The inverse Born series converges for the phaseless data considered when the scattering potential satisfies suitable smallness conditions.
    Required for the extension and radius-of-convergence analysis described for total-field data.

pith-pipeline@v0.9.0 · 5639 in / 1206 out tokens · 23006 ms · 2026-05-25T02:42:17.348217+00:00 · methodology

discussion (0)

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Reference graph

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