Persistent-Homology-Guided Topology Scanning of Qualitative Indicators for Acoustic Inverse Scattering

Jiaying Jia; Xiaomei Yang; Zhiliang Deng

arxiv: 2605.20673 · v1 · pith:RGOHPTHAnew · submitted 2026-05-20 · 🧮 math.NA · cs.NA· math-ph· math.MP

Persistent-Homology-Guided Topology Scanning of Qualitative Indicators for Acoustic Inverse Scattering

Xiaomei Yang , Jiaying Jia , Zhiliang Deng This is my paper

Pith reviewed 2026-05-21 03:03 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath-phmath.MP

keywords persistent homologyqualitative indicatorsacoustic inverse scatteringtopology detectionthreshold selectionBetti numberslinear sampling methodfactorization method

0 comments

The pith

Persistent homology of superlevel sets from any normalized qualitative indicator selects a topology-guided threshold for reconstructing acoustic scatterers.

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

The paper develops a postprocessing step that takes a gray-scale sampling indicator from methods like the linear sampling method or factorization method and turns it into a binary shape by choosing the right cutoff. Instead of picking the threshold by hand or by simple rules, the approach computes the persistent homology of the indicator's superlevel sets and extracts the zero- and one-dimensional features that track connected components and holes. These features are then used to estimate the unknown scatterer's topology and to pick the threshold that makes the Betti numbers of the reconstructed shape match the estimated topology while adding only mild geometric penalties. The construction stays inside the single-frequency setting and works with any normalized indicator or even a fusion of indicators.

Core claim

Given any normalized qualitative indicator, we scan the persistent homology of its superlevel sets and use the resulting zero- and one-dimensional persistent features to estimate or impose the topology of the unknown scatterer. A topology-guided threshold is then selected by minimizing a Betti-number discrepancy together with mild geometric penalties. The method is indicator-agnostic and includes an automatic topology detection rule based on persistence lifetimes and lifetime gaps.

What carries the argument

Persistent homology computed on the superlevel sets of the normalized indicator, which supplies zero- and one-dimensional persistent features used to estimate Betti numbers and to drive threshold selection by minimizing a discrepancy functional.

If this is right

The procedure supplies an automatic rule that detects the number of connected components and the number of holes without user-specified cutoffs.
It remains applicable to the linear sampling indicator, the factorization-method indicator, or any normalized linear combination of them.
The formulation uses only single-frequency data and therefore stays compatible with existing qualitative inverse-scattering pipelines.
Numerical tests show the method recovers correct topology for scatterers with multiple components or holes under moderate noise.

Where Pith is reading between the lines

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

The same superlevel-set persistence scan could be applied directly to other sampling indicators that appear in electromagnetic or elastic inverse scattering.
Because the threshold choice is driven by a global topology match rather than local intensity, the method may tolerate higher noise levels than intensity-based postprocessing.
An extension that incorporates multi-frequency indicators would allow the persistence diagram to be tracked across frequencies, potentially tightening the topology estimate.

Load-bearing premise

The persistent homology of the indicator's superlevel sets reliably reflects the true topology of the scatterer even when the indicator contains noise-induced artifacts or when the scatterer has nontrivial topology.

What would settle it

Generate a known scatterer with known topology, add increasing levels of noise to the data so that the resulting indicator contains clear artifacts, run the persistent-homology procedure, and check whether the selected threshold produces a binary shape whose topology matches the known scatterer; a mismatch at moderate noise levels would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.20673 by Jiaying Jia, Xiaomei Yang, Zhiliang Deng.

**Figure 2.** Figure 2: Fixed-level and PH-guided reconstructions for the LSM, FM, and fused indicators. [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: Leading finite persistence lifetimes for the three indicators. The dominant [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: Persistence diagrams of the distance-type indicators [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

read the original abstract

Qualitative methods such as the linear sampling method and the factorization method reconstruct acoustic scatterers through sampling indicators. In practice, these indicators are gray-scale fields on a prescribed sampling window and a binary obstacle shape is obtained only after thresholding. The choice of threshold is usually empirical and may be unstable when the indicator contains noise-induced artifacts or when the scatterer has nontrivial topology, such as multiple components or holes. This paper proposes a topology-aware postprocessing framework based on persistent homology. Given any normalized qualitative indicator, we scan the persistent homology of its superlevel sets and use the resulting zero- and one-dimensional persistent features to estimate or impose the topology of the unknown scatterer. A topology-guided threshold is then selected by minimizing a Betti-number discrepancy together with mild geometric penalties. The method is indicator-agnostic: it can be applied to the linear sampling indicator, the factorization-method indicator, or a normalized fusion of indicators. The main formulation is single-frequency and therefore remains close to the classical qualitative inverse scattering setting. We present the mathematical construction, an automatic topology detection rule based on persistence lifetimes and lifetime gaps, and a detailed algorithmic protocol for numerical implementation. Numerical tests verify that the proposed method is effective.

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.

Desk Editor's Note private letter to a colleague

The paper adds a persistent-homology post-processing step to pick thresholds on qualitative acoustic indicators so the output respects the scatterer's topology, but the reliability claim rests on an untested assumption about noise.

read the letter

The main takeaway is that this work gives a concrete way to replace empirical thresholding with a topology-aware rule that scans persistence on superlevel sets of any normalized indicator and then minimizes Betti-number mismatch plus light geometric penalties. It stays inside the single-frequency qualitative setting and applies equally to linear sampling, factorization, or fused indicators. That is the actual addition: an automatic detection rule based on lifetimes and gaps, plus a full algorithmic protocol for implementation. The numerical tests are presented as verification that the approach handles noise artifacts and nontrivial topologies better than plain thresholding. Those elements are useful and worth having on record for people who already run LSM or FM reconstructions. The formulation itself looks clean and indicator-agnostic, which keeps the overhead low. The soft spot is the central assumption that zero- and one-dimensional persistence of the indicator's superlevel sets will still recover the true Betti numbers when the field contains noise-induced local maxima or when the scatterer has holes or multiple components. No stability result is offered for that step, and the abstract supplies no quantitative error tables or baseline comparisons, so the practical gain remains hard to judge from the given material. The mild geometric penalties also introduce a few tunable weights that could affect outcomes on new data. This is aimed at researchers already working on qualitative inverse scattering who want a more systematic post-processing option. A reader comfortable with both inverse problems and basic persistent homology will get the most out of the construction and the protocol. The paper is coherent on its own terms and shows honest engagement with the practical limitation it targets, so it deserves a serious referee even if the robustness questions need to be addressed in revision. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a persistent-homology-based post-processing method to automatically select thresholds for qualitative indicators in acoustic inverse scattering. The approach involves computing the persistent homology of superlevel sets of the indicator field to extract topological features (Betti numbers in dimensions 0 and 1), then selecting a threshold that minimizes the discrepancy between the observed and expected Betti numbers, augmented by geometric penalty terms. The method is designed to be applicable to various indicators like the linear sampling method or factorization method, and includes an automatic detection rule using persistence lifetimes and gaps, along with numerical experiments to demonstrate its utility.

Significance. Should the proposed framework prove robust, it would offer a valuable tool for enhancing the reliability of qualitative inverse scattering reconstructions by addressing the often ad-hoc choice of thresholds, especially for scatterers with complex topologies or in the presence of noise. The indicator-agnostic nature and single-frequency formulation keep it aligned with classical settings, potentially broadening its applicability in practical imaging scenarios. The inclusion of a detailed algorithmic protocol is a positive aspect for reproducibility.

major comments (2)

Abstract: The assertion that 'Numerical tests verify that the proposed method is effective' provides no specific error metrics, data details, or comparison baselines, leaving the central empirical claim with limited verifiable support from the given text.
Mathematical construction: The framework rests on the assumption that zero- and one-dimensional persistence of superlevel sets of a normalized qualitative indicator reliably encodes the scatterer's Betti numbers even under noise or for nontrivial topology (β1 > 0); no stability result or analysis is supplied showing that lifetimes remain stable or that the Betti-discrepancy minimizer recovers the geometrically correct level set when spurious local maxima are present.

minor comments (2)

A brief reminder of basic persistent-homology definitions (e.g., superlevel-set filtration, lifetime gaps) would improve accessibility for readers primarily from the inverse-scattering community.
Figure captions should explicitly link displayed indicator fields to the corresponding persistence diagrams and the final selected threshold value.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive summary and for highlighting both the potential significance of the framework and the areas needing clarification. We address the two major comments point by point below.

read point-by-point responses

Referee: Abstract: The assertion that 'Numerical tests verify that the proposed method is effective' provides no specific error metrics, data details, or comparison baselines, leaving the central empirical claim with limited verifiable support from the given text.

Authors: We agree that the abstract statement would benefit from greater specificity. In the revised manuscript we will augment the abstract with a concise clause mentioning the quantitative metrics (Hausdorff distance to the true boundary and relative Betti-number error) together with the principal test configurations (single-frequency data, 5 % additive noise, and scatterers with β₁ = 1). This addition will supply the requested verifiable support without exceeding the abstract’s length limit. revision: yes
Referee: Mathematical construction: The framework rests on the assumption that zero- and one-dimensional persistence of superlevel sets of a normalized qualitative indicator reliably encodes the scatterer's Betti numbers even under noise or for nontrivial topology (β1 > 0); no stability result or analysis is supplied showing that lifetimes remain stable or that the Betti-discrepancy minimizer recovers the geometrically correct level set when spurious local maxima are present.

Authors: The referee correctly notes the absence of a rigorous stability theorem. The present work is deliberately empirical and algorithmic: it introduces an automatic detection rule based on persistence lifetimes and lifetime gaps, together with a Betti-discrepancy functional augmented by geometric penalties. Extensive numerical experiments (including noisy data and scatterers with holes) demonstrate that the selected threshold recovers the geometrically correct level set in practice. We will add a short subsection that explicitly states the modeling assumptions, cites the relevant persistence-stability literature, and discusses the conditions under which the method may fail. A full theoretical stability analysis lies outside the scope of this paper and would constitute a separate contribution. revision: partial

Circularity Check

0 steps flagged

No significant circularity: independent PH computation and optimization-based threshold selection

full rationale

The paper's chain proceeds by first computing zero- and one-dimensional persistent homology directly on the superlevel sets of any given normalized qualitative indicator (LSM, FM, or fusion), then applying an automatic detection rule using lifetimes and lifetime gaps to estimate Betti numbers, and finally selecting the threshold via minimization of Betti-number discrepancy plus mild geometric penalties. None of these steps reduces the claimed output to a fitted parameter or self-defined quantity by construction; the persistence features are extracted from the input indicator field without presupposing the scatterer's topology, and the minimization is an independent optimization step rather than a tautology. No load-bearing self-citations, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work are invoked in the provided formulation. The framework is therefore self-contained as a numerical post-processing algorithm.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on standard assumptions from persistent homology and optimization; no new physical entities are introduced.

free parameters (1)

weights for mild geometric penalties
Parameters balancing Betti-number discrepancy against geometric terms in threshold minimization; values not specified in abstract.

axioms (1)

domain assumption Persistent homology of superlevel sets captures the topological features (Betti numbers) of the underlying scatterer
Invoked when scanning zero- and one-dimensional persistent features to estimate or impose topology.

pith-pipeline@v0.9.0 · 5756 in / 1285 out tokens · 30863 ms · 2026-05-21T03:03:12.038735+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we scan the persistent homology of its superlevel sets and use the resulting zero- and one-dimensional persistent features to estimate or impose the topology of the unknown scatterer. A topology-guided threshold is then selected by minimizing a Betti-number discrepancy together with mild geometric penalties.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Adams, T

H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, and P. Shipman. Persistence images: a stable vector representation of persistent homology.Journal of Machine Learning Research, 18:1–35, 2017

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J. L. Buchanan, R. P. Gilbert, A. Wirgin, and Y. S. Xu.Marine Acoustics: Direct and Inverse Problems. SIAM, 2004

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Z. L. Deng, H. Y. Liu, X. F. Guan, Z. Y. Wang, and X. M. Yang. A persistent-homology- based Bayesian prior for potential coefficient reconstruction in an elliptic PDE.Statistics and Computing, accepted

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T. K. Dey and Y. Wang.Computational Topology for Data Analysis. Cambridge University Press, New York, 2022

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Edelsbrunner and J

H. Edelsbrunner and J. L. Harer.Computational Topology: An Introduction. American Mathematical Society, 2010

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H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete&Computational Geometry, 28:511–533, 2002

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T. Kaczynski, K. Mischaikow, and M. Mrozek.Computational Homology. Applied Mathematical Sciences. Springer New York, NY, 2004

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A. Kirsch and N. Grinberg.The Factorization Method for Inverse Problems. Oxford University Press, Oxford, 2008

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J. Z. Li and J. Zou. A direct sampling method for inverse scattering using far-field data.Inverse Problems and Imaging, 7(3):757–775, 2013

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J. Liu, X. D. Liu, and J. G. Sun. Extended sampling method for inverse elastic scattering problems using one incident wave.SIAM Journal on Imaging Sciences, 12(2):874–892, 2019

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Liu and J

J. Liu and J. G. Sun. Extended sampling method in inverse scattering.Inverse problems, 34(8):085007, 2018. 21

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Maria, J

C. Maria, J. D. Boissonnat, M. Glisse, and M. Yvinec.The GUDHI library: simplicial complexes and persistent homology, volume 8592 ofLecture Notes in Computer Science, pages 167–174. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014

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J. Ning, F. Han, and J. Zou. A direct sampling-based deep learning approach for inverse medium scattering problems.Inverse Problems, 40(015005):26pp, 2023

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C. S. Pun, S. X. Lee, and K. Xia. Persistent-homology-based machine learning: a survey and a comparative study.Artificial Intelligence Review, 55:5169–5213, 2022

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M. S. Zhdanov.Geophysical Inverse Theory and Regularization Problems, volume 36. Elsevier, 2002

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Zomorodian and G

A. Zomorodian and G. Carlsson. Computing persistent homology.Discrete&Computational Geometry, 33:249–274, 2005. 22

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[1] [1]

Adams, T

H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, and P. Shipman. Persistence images: a stable vector representation of persistent homology.Journal of Machine Learning Research, 18:1–35, 2017

work page 2017

[2] [2]

J. L. Buchanan, R. P. Gilbert, A. Wirgin, and Y. S. Xu.Marine Acoustics: Direct and Inverse Problems. SIAM, 2004

work page 2004

[3] [3]

Cohen-Steiner, H

D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Statility of persistence diagrams.Discrete& Computational Geometry, 37:103–120, 2007

work page 2007

[4] [4]

Colton and A

D. Colton and A. Kirsch. A simple method for solving inverse scattering problems in the resonance region.Inverse Problems, 12(4):383–393, 1996

work page 1996

[5] [5]

Colton and R

D. Colton and R. Kress.Inverse Acoustic and Electromagnetic Scattering Theory. Springer, 4 edition, 2019

work page 2019

[6] [6]

Z. L. Deng, H. Y. Liu, X. F. Guan, Z. Y. Wang, and X. M. Yang. A persistent-homology- based Bayesian prior for potential coefficient reconstruction in an elliptic PDE.Statistics and Computing, accepted

work page

[7] [7]

T. K. Dey and Y. Wang.Computational Topology for Data Analysis. Cambridge University Press, New York, 2022

work page 2022

[8] [8]

Edelsbrunner and J

H. Edelsbrunner and J. L. Harer.Computational Topology: An Introduction. American Mathematical Society, 2010

work page 2010

[9] [9]

Edelsbrunner, D

H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete&Computational Geometry, 28:511–533, 2002

work page 2002

[10] [10]

Fern´ andez-Funentes, C

X. Fern´ andez-Funentes, C. Mera, A. G´ omez, and I. Vidal-Franco. Towards a fast and accurate eit inverse problem solver: a machine learning approach.Electronics, 7(12), 2018

work page 2018

[11] [11]

Kaczynski, K

T. Kaczynski, K. Mischaikow, and M. Mrozek.Computational Homology. Applied Mathematical Sciences. Springer New York, NY, 2004

work page 2004

[12] [12]

A. Kirsch. Characterization of the shape of a scattering obstacle using the spectral data of the far field operator.Inverse Problems, 14(6):1489–1512, 1998

work page 1998

[13] [13]

Kirsch.An Introduction to the Mathematical Theory of Inverse Problems, volume 120

A. Kirsch.An Introduction to the Mathematical Theory of Inverse Problems, volume 120. Springer, 2011

work page 2011

[14] [14]

Kirsch and N

A. Kirsch and N. Grinberg.The Factorization Method for Inverse Problems. Oxford University Press, Oxford, 2008

work page 2008

[15] [15]

J. Z. Li and J. Zou. A direct sampling method for inverse scattering using far-field data.Inverse Problems and Imaging, 7(3):757–775, 2013

work page 2013

[16] [16]

J. Liu, X. D. Liu, and J. G. Sun. Extended sampling method for inverse elastic scattering problems using one incident wave.SIAM Journal on Imaging Sciences, 12(2):874–892, 2019

work page 2019

[17] [17]

Liu and J

J. Liu and J. G. Sun. Extended sampling method in inverse scattering.Inverse problems, 34(8):085007, 2018. 21

work page 2018

[18] [18]

Maria, J

C. Maria, J. D. Boissonnat, M. Glisse, and M. Yvinec.The GUDHI library: simplicial complexes and persistent homology, volume 8592 ofLecture Notes in Computer Science, pages 167–174. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014

work page 2014

[19] [19]

J. Ning, F. Han, and J. Zou. A direct sampling-based deep learning approach for inverse medium scattering problems.Inverse Problems, 40(015005):26pp, 2023

work page 2023

[20] [20]

C. S. Pun, S. X. Lee, and K. Xia. Persistent-homology-based machine learning: a survey and a comparative study.Artificial Intelligence Review, 55:5169–5213, 2022

work page 2022

[21] [21]

M. S. Zhdanov.Geophysical Inverse Theory and Regularization Problems, volume 36. Elsevier, 2002

work page 2002

[22] [22]

Zomorodian and G

A. Zomorodian and G. Carlsson. Computing persistent homology.Discrete&Computational Geometry, 33:249–274, 2005. 22

work page 2005