Inverse obstacle scattering regularized by the tangent-point energy

TL;DR

This work develops a regularization framework for 3D inverse obstacle scattering by employing the tangent-point energy ${\mathcal{E}}_p$ as a generalized Tikhonov penalty on surface embeddings. It combines a boundary-element forward model, a covariant Sobolev-metric preconditioner, and an iteratively regularized Gauss–Newton method to update surfaces while preserving embedding/isotopy class. The authors establish existence and regularization properties, derive smoothness and differentiation of the energy, and implement a scalable numerical pipeline using tree-accelerated energy evaluation and GPU-accelerated boundary operators. Numerical experiments on complex geometries, varying wavelengths, and reduced data scenarios demonstrate high-quality reconstructions even under substantial noise, underscoring the method’s robustness and practical potential for challenging inverse scattering problems.

Abstract

We employ the so-called tangent-point energy as Tikhonov regularizer for ill-conditioned inverse scattering problems in 3D. The tangent-point energy is a self-avoiding functional on the space of embedded surfaces that also penalizes surface roughness. Moreover, it features nice compactness and continuity properties. These allow us to show the well-posedness of the regularized problems and the convergence of the regularized solutions to the true solution in the limit of vanishing noise level. We also provide a reconstruction algorithm of iteratively regularized Gauss-Newton type. Our numerical experiments demonstrate that our method is numerically feasible and effective in producing reconstructions of unprecedented quality.

Figures (13)

• Figure 1: (a) The original Stanford bunny together with the sphere we used as initial guess. The synthetic far field data was perturbed by $1\%$ Gaussian white noise. (b) A rough reconstruction, shaded by the signed distance field of the true obstacle. We used wavelengths $\lambda \in \braces{1, 2}$, each in $16$ wave directions, roughly evenly distributed over the unit sphere. (c) The final reconstruction (16 waves with $\lambda = 1/2$), also shaded by the signed distance field of the true obstacle. This state was reached from (b) in $32$ Gauss--Newton steps with discrepancy parameter $\tau=2$ (see \ref{['sec:IRGNM']} for the meaning of $\tau$).
• Figure 2: (a) The surface Blub as obstacle for the plane wave $u_{\mathrm{i}}$, incoming from the right-hand side. (b) The scattered wave $u_{\mathrm{s}}$ of the solution to the boundary value problem \ref{['eq:bvp']}. (c) The total solution $u$ of the problem. Notice that it vanishes on the boundary of the obstacle, correctly modelling the sound-absorbing behavior.
• Figure 3: Splitting of the triangle for the semi-analytic quadrature.
• Figure 4: Visualization of the matrix-matrix product. Each work group handles a block of size $W \times W$ of $A_{\kappa_\ell}$ and a block of size $W \times D$ of $B_{\kappa_\ell}$, where $W$ is the work group size and $D$ is the number of incident wave directions.
• Figure 5: A log-log plot of the relative max error of the operator images against the average edge length $h$ for a collection of spheres. $F$ denotes the discretized boundary-to-far field map.

Theorems & Definitions (41)

• Theorem 1: Geometric Sobolev–Morrey embedding
• Theorem 3.1: Rigidity
• Remark 1
• Theorem 3.2: Compactness
• Definition 1
• Theorem 3.3: Lower semicontinuity
• Lemma 1
• Proof 1
• Theorem 3.4
• Lemma 2