Xiaomei Yang, Jiaying Jia, Zhiliang Deng
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.