When topological derivatives met regularized Gauss-Newton iterations in holographic 3D 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 Fréchet derivatives are determined.

Bayesian approach to inverse scattering with topological priors

“…Assuming material properties to be characterized by constant parameters, the proposed framework can be adopted by including a few additional parameters. To consider more general spatially variable material properties, one could combine these methods with those developed in [6,54] and implement coupled BEM-FEM or spectral-FEM solvers as in [12,13]. The methods would extend to wave fields governed by systems different from Helmholtz equations, provided characterizations for the derivatives and adequate solvers for the boundary value problems involved are available.…”

“…, keeping the previous notations and following [12,13]: 17) where u and p solve the forward and adjoint problems (1) and (A.18) ∆p + κ 2 e p = N j=1 χ(x j )δ x j in R 2 \ Ω i , ∆p + κ 2 i p = 0 in Ω i , p − − p + = 0 on ∂Ω i , β∂ n p − − ∂ n p + = 0 on ∂Ω i , lim with r = |x|. This expression agrees with that established in [10,12] when V = V n n. The same proofs hold for general fields V keeping track of the terms involving tangential components and the transmission boundary conditions.…”

“…In a light holography set-up, the general framework is similar [12,13], assuming we use polarized light in the presence of few well separated objects. Then, the equations governing the amplitude field can be approximated by (1) and the wavenumbers are again given by (12), c and ν representing light wavespeed and frequencies.…”

Single‐Particle Characterization by Elastic Light Scattering

“…More recently Carpio et al. [ 234,235 ] developed another holographic reconstruction algorithm based on the topological derivatives of the electromagnetic field. First, they used a scalar forward model and assumed the RI close to 1 to build a two‐step algorithm, which determines support of the scatterer (where the RI is different from 1), and then fits the RI using a spherical approximation to the particle or its components.…”

“…[ 234 ] Next, they improved the algorithm for 3D shape reconstruction, using the topological and Frechet (shape) derivatives of the cost (residual) functional based on holography measurements. [ 235 ] This algorithm can determine both the number of components and the shape of each one, assuming that the RI is constant and known. Moreover, it describes, how any light‐scattering method can be adapted to calculate corresponding derivatives, by using a specially formed incident field.…”