A surrogate-Bayesian algorithm for scatterer shape identification from phaseless data

Abstract

This work addresses the reconstruction of a scatterer's shape from phaseless far field-intensity data arising from multiple incident waves interacting with the scatterer. We formulate the reconstruction as a statistical inverse scattering problem and adopt a Bayesian inference framework, which can readily be used to compute statistical moments for quantification of uncertainties in the shape reconstruction that arise from noise in the data due to measurement constraints. The shape of the scatterer is represented by a spline-based prior, with Bayesian parameters defined at the spline's knots. To efficiently evaluate the Bayesian likelihood across thousands of sampling points, we develop the intensity property inspired neural network (IPINN) surrogate. This surrogate incorporates the Helmholtz equation in the unbounded domain, exterior to each sampled scatterer, along with the radiation condition at infinity, enabling fast and accurate simulation of the acoustic far-field intensity. Importantly, the IPINN surrogate is trained independently of the observed data and requires only a single incident wave for training. We demonstrate that this surrogate approach yields a speed-up of several orders of magnitude. The resulting IPINN-Bayesian framework offers an efficient solution for shape reconstruction in unbounded domains with multiple incident wave boundary conditions, while exactly enforcing the radiation condition. Numerical experiments confirm the efficiency and effectiveness of the proposed algorithm.

Figures (9)

• Figure 1: Visualization of $1000$ samples of the prior space (shaded grey) with $N_\text{spline}=8$ and $\xi_i \sim \mathcal{U}[-0.5,0.5]$ for $i=1,\dots,8$, so that $- \log R_0 = \log R = 0.5$. One sample is highlighted in black with its knot values marked.
• Figure 2: Effect of the post-processing alignment step in shape reconstruction. (Top left) 6 random reconstruction samples from the posterior distribution and their associated center of mass (plus markers), (top right) corresponding statistical inference for all samples. (Bottom left) The same 6 samples after alignment using (\ref{['eq: recentering']}) and (bottom right) corresponding confidence intervals. The parameters are given in Figure \ref{['fig:Nspline=Nobs=12 reconstructions']}.
• Figure 3: Histograms showing normalized frequency of the relative error (\ref{['eq: n vs n+5 rel error']}) for $1000$ random shapes $D(\boldsymbol{\xi}^{(i)})$ with $\xi_i \sim \mathcal{U}[-0.5,0.5]$ and $N_\text{spline}=N_\text{obs}=12$ at $k=\pi$ (left) and $k=2\pi$ (right) both with Nyström discretization parameter $n=100$.
• Figure 4: Error histogram of the NN predictions for $k=\pi$ (left) and $k=2\pi$ (right) surrogates used for reconstructions. Training (blue), validation (green), and test (red, dashed) sets show consistent zero-centered distributions with similar low RMSE values across all datasets.
• Figure 5: Shape reconstructions from the posterior distribution across the catalogue of shapes at $k=\pi$ with $N_\text{spline}=N_\text{inc}=N_\text{obs}=12$ and $2\%$ noise ($\hat{\sigma}=0.02$). The black lines describe the 'true shapes', the red line represents the mean-shape, and the red-shaded area gives the associated mean $\pm 2\sigma$ confidence interval.