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

TL;DR

This paper introduces a topology-aware postprocessing method using persistent homology to improve qualitative acoustic inverse scattering, especially in noisy or complex topologies.

Contribution

It proposes a novel, indicator-agnostic framework that estimates and enforces the scatterer's topology via persistent homology, enhancing stability and accuracy.

Findings

Effective in noisy conditions

Accurately detects topology of scatterers

Compatible with multiple indicators

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.