Positive Semidefinite Relaxations for Imaging Science

This thesis proposes theoretical and algorithmic advances for positive semi-definite relaxations and their applications in data science. These so-called Lasserre’s hierarchies allow one to solve super-resolution problems without resorting to spatial discretization, by replacing measures with their trigonometric moments. However, they require the resolution of large convex optimization problems. The contributions of this thesis show how to scale these methods by exploiting certain properties and invariances of imaging problems. We first propose a new approach for the recovery of continuous measures from a finite number of moments, based on the approximate joint-diagonalization of a few non-Hermitian matrices. Then, we study the super-resolution problem, and its approximation by Lasserre’s hierarchy. A preliminary step consisting in the spectral projection of the forward operator makes this approximation possible, and further allows a fast implementation, the Fourier-based Frank Wolfe (FFW), taking advantage of the convolutional and low-rank structure of the involved matrices. We apply our method on fluorescence microscopy data. Finally, combining the recovery of continuous measures with the fast implementation of FFW, we employ Lasserre’s hierarchies to approximate the optimal transport between measures, which can also be coupled with the Blasso.