Solving Inverse Obstacle Scattering Problem with Latent Surface Representations

TL;DR

This work addresses the inverse obstacle scattering problem by introducing a trained latent surface prior via DeepSDF to represent obstacle boundaries as zero level sets of a differentiable generator. By deriving a shape-derivative-based gradient with respect to the latent variables and employing ADAM with a projection step onto the latent manifold, the method achieves fast, robust reconstructions from far-field data, including backscattering and phaseless measurements. Theoretical convergence guarantees for the gradient-based scheme are provided, and numerical experiments on airplane and car geometries demonstrate strong noise robustness (up to 40% relative noise) and efficient convergence. The approach significantly reduces the optimization dimensionality and leverages expressive latent priors to overcome ISP ill-posedness, with practical implications for remote sensing, nondestructive evaluation, and related imaging tasks.

Abstract

We propose a novel iterative numerical method to solve the three-dimensional inverse obstacle scattering problem of recovering the shape of the obstacle from far-field measurements. To address the inherent ill-posed nature of the inverse problem, we advocate the use of a trained latent representation of surfaces as the generative prior. This prior enjoys excellent expressivity within the given class of shapes, and meanwhile, the latent dimensionality is low, which greatly facilitates the computation. Thus, the admissible manifold of surfaces is realistic and the resulting optimization problem is less ill-posed. We employ the shape derivative to evolve the latent surface representation, by minimizing the loss, and we provide a local convergence analysis of a gradient descent type algorithm to a stationary point of the loss. We present several numerical examples, including also backscattered and phaseless data, to showcase the effectiveness of the proposed algorithm.

Figures (16)

• Figure 1: A schematic illustration of DeepSDF park2019deepsdf.
• Figure 2: The interpolation between the first and second columns in the learned shape latent space, shown in the third column, from two view angles.
• Figure 3: Example triangular mesh with 1190 vertices and 2376 faces.
• Figure 4: The convergence of the optimization algorithm for Scenario 1: the intermediate reconstruction at iterations: 0, 26, 53, 80, 107, 134, and 161 (ordered from left to right and from top to bottom), and the last plot denotes the exact target.
• Figure 5: The reconstructions for Scenario 1 with exact data: the left, middle, and right columns refer to the initial, the optimal recovery, and the ground truth target, respectively.

Theorems & Definitions (15)

• Theorem 3.1
• proof
• Theorem 4.1
• Lemma 4.2
• Lemma 4.3
• proof
• Lemma 4.4
• proof
• proof : Proof of Theorem \ref{['hessian']}
• Theorem 4.5