A Statistical Imaging Framework for Inverse Obstacle Scattering and Its Resolution Analysis

We develop a general statistical framework for inverse obstacle imaging and resolution analysis from far-field second moments under random illumination. The regularized $Q$-statistic is introduced and utilized to measure the minimal statistically weighted cost of synthesizing the test far field associated with a sampling point. Its reciprocal provides an imaging indicator. Under uniform full-aperture plane-wave illumination, we relate this indicator to several classical sampling indicators and derive their statistical counterparts for general random illumination. For the resolution analysis, separating the effects of radiation and noise from the statistics of induced boundary sources, we measure a correlation-free resolution limit by the one-sided half-height width of the boundary transition. In the regime of fixed wavenumber $k$ with the noise-calibrated regularization level $ρ_τ\to 0$, we establish a sharp asymptotic law for this width for simply connected domains with analytic boundaries. The width scales as (h_Γ(x)\log\log(1/ρ_τ)/\log(1/ρ_τ)), with $h_Γ(x)$ a geometric prefactor along the boundary determining both the boundary image contrast and the local transition width. For general boundary-source statistics, we examine how the source mean, variance, and correlation affect imaging quality. Finally, in representative scattering configurations, we provide qualitative interpretations of some typical imaging phenomena through the statistical structure of their induced boundary sources. Numerical examples illustrate these mechanisms and indicate that practical resolution is governed jointly by source statistics, radiation, the noise level, rather than by wavelength alone.

Source: arXiv:2610.28553v1 - http://arxiv.org/abs/2610.28553v1 PDF: https://arxiv.org/pdf/2610.28553v1 Original Link: http://arxiv.org/abs/2610.28553v1