# When topological derivatives met regularized Gauss-Newton iterations in holographic 3D imaging

## TL;DR

This paper introduces an automatic 3D electromagnetic imaging algorithm combining topological derivatives and regularized Gauss-Newton iterations, enabling accurate nanoscale reconstructions from holograms with fewer detectors.

## Contribution

It develops a novel method integrating topological derivatives with Gauss-Newton iterations for improved 3D holographic imaging, allowing automatic object number updates during optimization.

## Key findings

- Achieves fast, accurate 3D shape reconstructions with nanoscale resolution.
- Handles multiple components and non-convex geometries effectively.
- Requires fewer detectors for reliable imaging.

## Abstract

We propose an automatic algorithm for 3D inverse electromagnetic scattering based on the combination of topological derivatives and regularized Gauss-Newton iterations. The algorithm is adapted to decoding digital holograms. A hologram is a two-dimensional light interference pattern that encodes information about three-dimensional shapes and their optical properties. The formation of the hologram is modeled using Maxwell theory for light scattering by particles. We then seek shapes optimizing error functionals which measure the deviation from the recorded holograms. Their topological derivatives provide initial guesses of the objects. Next, we correct these predictions by regularized Gauss-Newton techniques. In contrast to standard Gauss-Newton methods, in our implementation the number of objects can be automatically updated during the iterative procedure by new topological derivative computations. We show that the combined use of topological derivative based optimization and iteratively regularized Gauss-Newton methods produces fast and accurate descriptions of the geometry of objects formed by multiple components with nanoscale resolution, even for a small number of detectors and non convex components aligned in the incidence direction. The method could be applied in general imaging set-ups involving other waves (microwave imaging, elastography...) provided closed-form expressions for the topological and Frechet derivatives are determined.