## Section: New Results

### Phase retrieval for imaging problems

Participants : Fajwel Fogel [correspondent] , Irène Waldspurger, Alexandre d'Aspremont.

In [29] we study convex relaxation algorithms for phase retrieval on imaging problems. We show that structural assumptions on the signal and the observations, such as sparsity, smoothness or positivity, can be exploited to both speed-up convergence and improve recovery performance. We detail experimental results in molecular imaging problems simulated from PDB data.

Phase retrieval seeks to reconstruct a complex signal, given a number of observations on the *magnitude* of linear measurements, i.e. solve

in the variable $x$, where $A$ and $b$. This problem has direct applications in X-ray and crystallography imaging, diffraction imaging, Fourier optics or microscopy for example, in problems where physical limitations mean detectors usually capture the intensity of observations but cannot recover their phase. In this project, we focus on problems arising in diffraction imaging, where $A$ is usually a Fourier transform, often composed with one or multiple masks (a technique sometimes called ptychography). The Fourier structure, through the FFT, often considerably speeds up basic linear operations, which allows us to solve large scale convex relaxations on realistically large imaging problems. We also observe that in most of the imaging problems we consider, the Fourier transform is very sparse, with known support (we lose the phase but observe the magnitude of Fourier coefficients), which allows us to considerably reduce the size of our convex phase retrieval relaxations.