## Thursday, July 04, 2013

### CO4OI: Convex Optimisation for Optical Interferometry -implementation-

Yves Wiaux just sent me the following:

Hi Igor,
We have a new submission on partial phase retrieval in optical interferometry (nonlinear bispectrum measurements) adopting a convex order-3 tensor recovery perspective. ...Code will be available in a few days on the BASP Github page highlighted in the article.
Regards
Yves

Uh! making a nonlinear sensing scheme linear "by lifting the image model from a vector to a tensor formulation", that sound eminently interesting in light of the increasingly different approaches to imaging (see Sunday Morning Insight: A Quick Panorama of Sensing from Direct Imaging to Machine Learning). To see right away the improvements of the paper, here is some definition for NM and AM:

The target approach is a convex nuclear minimisation (NM) scheme with built-in supersymmetry. For comparison, and in the line of the current state of the art, we also study a nonconvex alternate minimisation (AM) approach where the rank-1 constraint is built-in

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. We propose a convex approach for tensor recovery with built-in supersymmetry, and regularising the inverse problem through a nuclear norm relaxation of a low-rank constraint. For comparison, and in the line of the current state of the art, we also study a nonlinear nonconvex approach. Keeping our tensor perspective, the problem is formulated for the tensor product of 3 vectors, where supersymmetry is relaxed while the rank-1 constraint is built-in. Linear convex minimisation problems are solved alternately and iteratively for these vectors. Simulation results show that the convex scheme provides significantly superior and more stable imaging quality than the nonconvex approach, both for randomly generated signals and realistic images. Code and test data are available at this https URL

The implementation will be located here shortly:

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