Bayesian approach for inverse obstacle scattering with Poisson data

Xiaomei Yang; Zhiliang Deng

arxiv: 1907.03955 · v1 · pith:PGD2JJ6Lnew · submitted 2019-07-09 · 🧮 math.NA · cs.NA· math.ST· stat.TH

Bayesian approach for inverse obstacle scattering with Poisson data

Xiaomei Yang , Zhiliang Deng This is my paper

Pith reviewed 2026-05-25 00:32 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.STstat.TH

keywords Bayesian inversioninverse obstacle scatteringPoisson datahybrid priortotal variationangular parameterizationacoustic scatteringwell-posedness

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The pith

A hybrid Gaussian-total variation prior on angular obstacle parameterization yields a well-posed posterior for Bayesian inversion of Poisson acoustic scattering data.

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

The paper applies Bayesian inversion to recover the shape of an unknown obstacle from acoustic scattering measurements that follow a Poisson distribution. It parameterizes the obstacle boundary in angular coordinates and places a hybrid prior on this parameterization that combines a Gaussian measure with a total-variation measure. The authors prove that this choice yields a well-posed posterior distribution. Numerical tests then demonstrate that the resulting algorithm can reconstruct the obstacle geometry under the given noise model.

Core claim

We consider an acoustic obstacle reconstruction problem with Poisson data tackled in the Bayesian inversion framework. The unknown obstacle is parameterized in angular form. Under the assumption that the unknown satisfies both a Gaussian prior and a total variation prior, the hybrid prior is used and the well-posedness of the posterior distribution is discussed. Numerical examples verify the effectiveness of the proposed algorithm.

What carries the argument

The hybrid prior formed as the product of a Gaussian measure and a total-variation measure on the angular parameterization of the obstacle boundary.

If this is right

The posterior distribution exists and is stable with respect to the data, enabling consistent Bayesian inference.
The reconstruction algorithm can be applied directly to acoustic scattering experiments that produce Poisson-distributed count data.
The hybrid prior regularizes the inverse problem sufficiently for the angular parameterization to produce usable reconstructions.
Numerical performance observed in the examples follows from the well-posedness result.

Where Pith is reading between the lines

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

The same hybrid prior construction could be tested on other inverse problems whose data are best modeled by Poisson statistics rather than additive Gaussian noise.
The angular parameterization implicitly restricts the method to star-shaped obstacles; extensions to more general shapes would require a different representation.
If the total-variation component dominates, the reconstructions may favor piecewise-constant boundaries even when the true obstacle is smooth.

Load-bearing premise

The unknown obstacle geometry can be adequately represented by an angular parameterization whose prior is the product of Gaussian and total-variation measures.

What would settle it

A concrete counterexample in which the posterior fails to be a well-defined probability measure on the parameter space, or in which repeated numerical reconstructions from simulated Poisson data systematically fail to recover known star-shaped obstacles, would falsify the central claim.

read the original abstract

We consider an acoustic obstacle reconstruction problem with Poisson data. Due to the stochastic nature of the data, we tackle this problem in the framework of Bayesian inversion. The unknown obstacle is parameterized in its angular form. The prior for the parameterized unknown plays key role in the Bayes reconstruction algorithm. The most popular used prior is the Gaussian. Under the Gaussian prior assumption, we further suppose that the unknown satisfies the total variation prior. With the hybrid prior, the well-posedness of the posterior distribution is discussed. The numerical examples verify the effectiveness of the proposed algorithm.

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

This applies standard Bayesian inversion to Poisson-noise acoustic scattering via angular parameterization and a hybrid Gaussian-TV prior, with numerical checks but limited scope.

read the letter

The core of the paper is a Bayesian setup for recovering star-shaped obstacles from far-field data that follows a Poisson distribution. They parameterize the boundary by a radial function r(θ), put a hybrid prior on it that combines a Gaussian measure with total variation, prove the posterior is well-posed under that prior, and run some numerical examples that recover the shape from simulated Poisson data. That is the actual contribution: an incremental extension of existing Bayesian machinery to this noise model and this prior combination in the scattering setting. The numerics appear to confirm that the hybrid prior can produce reasonable reconstructions on the chosen test cases. The well-posedness discussion is the part that goes beyond pure application. The modeling choice is explicit and the Poisson likelihood is handled directly rather than approximated by Gaussian noise. The soft spots are straightforward. Everything rests on the obstacle being star-shaped so that the angular parameterization is valid; nothing in the abstract or stress-test note indicates they handle non-star-shaped scatterers. The continuity of the forward map from the hybrid space into the Poisson likelihood is asserted but the abstract supplies no derivation or error bounds, so it is not possible to judge how tight the argument is. If the TV term destroys the regularity the Gaussian part is meant to provide, or if the data deviate from Poisson, the posterior guarantees would not transfer. The numerical examples are only as convincing as the test cases chosen. This is for specialists already working on Bayesian inverse scattering who want to see how a hybrid prior behaves with Poisson data. A reader outside that niche will not find a new framework or broad methodological advance. It is worth sending to peer review because the topic is well-defined, the prior choice is reproducible, and the claims are falsifiable once the full proofs and data details are examined. A referee can check whether the continuity argument holds and whether the numerics are robust beyond the star-shaped examples.

Referee Report

3 major / 3 minor

Summary. The manuscript develops a Bayesian inversion framework for acoustic obstacle scattering problems in which the measured far-field data are modeled as Poisson random variables. The unknown obstacle is represented via an angular (radial) parameterization r(θ), and a hybrid prior that combines a Gaussian measure with a total-variation penalty is introduced. The authors establish well-posedness of the resulting posterior distribution and present numerical examples to illustrate the reconstruction algorithm.

Significance. If the well-posedness argument is complete and the forward map is shown to be continuous from the hybrid space into the Poisson likelihood, the work supplies a theoretically grounded method for shape recovery under realistic count-data noise. The hybrid prior construction is a concrete technical contribution that could be reused in other inverse-scattering settings where both smoothness and edge preservation are required.

major comments (3)

[§4] §4 (well-posedness of the posterior): the argument that the hybrid Gaussian-TV prior yields a proper probability measure requires explicit verification that the Poisson negative-log-likelihood is lower semi-continuous with respect to the total-variation seminorm; without this estimate the claim that the posterior is well-defined rests on an unproven continuity step.
[§2] §2 (angular parameterization): every admissible obstacle is required to be star-shaped so that r(θ) is single-valued; the manuscript supplies no analysis or counter-example showing what happens when this modeling assumption fails, which directly limits the scope of both the well-posedness theorem and the numerical claims.
[Numerical examples] Numerical examples section: the reported reconstructions use only star-shaped test objects and Poisson noise generated from the forward model; no quantitative comparison (e.g., L² or Hausdorff error) against a pure Gaussian prior or against non-star-shaped geometries is given, so the practical benefit of the TV component cannot be assessed.

minor comments (3)

[Abstract] Abstract, line 3: 'The most popular used prior' is grammatically incorrect; 'The most commonly used prior' would be clearer.
[§3] Notation for the hybrid prior measure is introduced without an explicit product-space definition or reference to the precise function space (e.g., H¹ ∩ BV); this makes the subsequent well-posedness statements harder to parse.
[Figures] Figure captions do not indicate whether the displayed reconstructions are single realizations or averages over multiple Poisson draws; given the stochastic data model this information is needed for reproducibility.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We appreciate the referee's detailed review and constructive comments on our manuscript. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

Referee: [§4] §4 (well-posedness of the posterior): the argument that the hybrid Gaussian-TV prior yields a proper probability measure requires explicit verification that the Poisson negative-log-likelihood is lower semi-continuous with respect to the total-variation seminorm; without this estimate the claim that the posterior is well-defined rests on an unproven continuity step.

Authors: We agree that the lower semi-continuity of the Poisson negative-log-likelihood with respect to the total-variation seminorm requires an explicit proof to complete the well-posedness argument. In the revised manuscript we will insert a new lemma establishing this continuity property together with its proof. revision: yes
Referee: [§2] §2 (angular parameterization): every admissible obstacle is required to be star-shaped so that r(θ) is single-valued; the manuscript supplies no analysis or counter-example showing what happens when this modeling assumption fails, which directly limits the scope of both the well-posedness theorem and the numerical claims.

Authors: The angular parameterization is deliberately chosen to represent star-shaped obstacles, a standard modeling assumption that guarantees a single-valued radial function. We will add a clarifying remark in Section 2 that explicitly states the scope of this assumption and its consequences for the theorems and experiments. revision: partial
Referee: Numerical examples section: the reported reconstructions use only star-shaped test objects and Poisson noise generated from the forward model; no quantitative comparison (e.g., L² or Hausdorff error) against a pure Gaussian prior or against non-star-shaped geometries is given, so the practical benefit of the TV component cannot be assessed.

Authors: We agree that quantitative comparisons would strengthen the numerical section. We will add L² and Hausdorff error tables comparing the hybrid prior against a pure Gaussian prior on the same star-shaped test objects. Comparisons involving non-star-shaped geometries lie outside the present parameterization and are therefore not included. revision: partial

standing simulated objections not resolved

The angular parameterization fundamentally requires star-shaped obstacles; a full analysis or counter-examples for non-star-shaped geometries cannot be supplied within the current framework.

Circularity Check

0 steps flagged

No circularity; standard Bayesian well-posedness from hybrid prior on angular parameterization.

full rationale

The derivation applies the established Bayesian inversion framework to Poisson data, with the obstacle represented via angular parameterization and a hybrid Gaussian-TV prior. Well-posedness of the posterior follows from continuity properties of the forward scattering map and the chosen prior measure, without any equation reducing a claimed prediction to a fitted input by construction. No self-citation chains, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are present. Numerical examples function as verification of the algorithm rather than as the source of the theoretical claims. The modeling assumptions (star-shaped obstacles, Poisson statistics) are explicit but do not create circularity in the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or non-standard axioms are stated.

axioms (1)

domain assumption The posterior distribution is well-posed under the hybrid Gaussian-total variation prior on the angular parameterization.
Invoked when the authors state that well-posedness is discussed.

pith-pipeline@v0.9.0 · 5616 in / 1084 out tokens · 23597 ms · 2026-05-25T00:32:44.749852+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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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

The unknown obstacle is parameterized in its angular form... hybrid prior... well-posedness of the posterior distribution is discussed.
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

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

starlike... ∂D = q(t)[cos(t), sin(t)]^T

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