Tensor optimisation for optical-interferometric imaging

TL;DR

This paper introduces two novel tensor-based methods for optical interferometric image reconstruction, reformulating the problem as linear convex and nonlinear nonconvex optimizations, and compares their effectiveness through simulations.

Contribution

It proposes a linear convex tensor recovery method with nuclear norm regularization and a nonlinear nonconvex approach with rank-1 constraints, both tailored for optical interferometry.

Findings

The convex approach effectively recovers images with supersymmetry constraints.

The nonconvex approach uses iterative minimization of linear subproblems.

Comparative analysis shows strengths and limitations of both methods.

Abstract

Image recovery in optical interferometry is an ill-posed nonlinear inverse problem arising from incomplete power spectrum and bispectrum measurements. We reformulate this nonlin- ear problem as a linear problem for the supersymmetric rank-1 order-3 tensor formed by the tensor product of the vector representing the image under scrutiny with itself. On one hand, we propose a linear convex approach for tensor recovery with built-in supersymmetry, and regularising the inverse problem through a nuclear norm relaxation of a low-rank constraint. On the other hand, we also study a nonlinear nonconvex approach with built-in rank-1 con- straint but where supersymmetry is relaxed, formulating the problem for the tensor product of 3 vectors. In this second approach, only linear convex minimisation subproblems are how- ever solved, alternately and iteratively for the 3 vectors. We provide a comparative analysis of these two novel approaches through numerical simulations on small-size images.