Tensor optimization for optical-interferometric imaging

Image recovery in optical interferometry is an ill-posed non-linear inverse problem arising from incomplete power spectrum and bispectrum measurements. We reformulate this nonlinear 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 regularizing the inverse problem through a nuclear norm relaxation of a low-rank constraint. On the other hand, we also study a non-linear non-convex approach with a built-in rank-1 constraint but where supersymmetry is relaxed, formulating the problem for the tensor product of three vectors. In this second approach, only linear convex minimization subproblems are, however, solved, alternately and iteratively for the three vectors. We provide a comparative analysis of these two novel approaches through numerical simulations on small-size images. © 2013 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.