The problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval $[-\Omega, \Omega]$ is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length $\frac{\pi}{\Omega}$. In this talk, we present a super-resolution algorithm, called Iterative Focusing-localization and Filtering (IFF), to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm requires no prior information about the source numbers and allows for a subsampling strategy that can circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods. In the talk, we will also discuss the theoretical results of the methods behind the algorithm. The derived results imply a phase transition phenomenon. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit as well as the phase transition phenomenon predicted by the theoretical analysis.

Blind super-resolution is the problem of estimating high-resolution information about a signal from its low-resolution measurements when the point spread functions (PSFs) are unknown. It is a common problem in many scientific and engineering research areas, such as machine learning, signal processing, and computer vision. Blind super-resolution can be cast as a low-rank matrix recovery problem by exploiting the inherent simplicity of the signal and the low-dimensional structure of the PSFs.

In this talk, we will discuss the low-rank matrix recovery problem for blind super-resolution of point sources. The target matrices associated with these problems are not only low rank but also highly structured. Convex approaches are proposed for the corresponding low-rank matrix recovery problems. Theoretical guarantees are established showing that near-optimal sample complexity is sufficient for successful recovery.

The behaviour of sparse regularization using the Lasso method is well understood when dealing with discretized linear models. However, the behaviour of Lasso is poor when dealing with models with very large parameter spaces and exact localisation of the sparse support is often not possible due to discretization (gridding) issues. We introduced a new optimization problem known as the super-resolved Lasso, by considering a higher order expansion of the continuous operator, we show that we can precisely recover the support when the 'true' signal lies up to a fraction off the grid. This is joint work with Gabriel Peyre.

In this work, we propose to apply and adapt known results on convex variational methods for inverse problems to the inverse scattering problem. We rely on approximations to circumvent its nonlinearity, and discuss recovery guarantees and numerical methods.