We study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. When the point sources are located in several clumps, we provide explicit upper bounds for MUSIC and ESPRIT in terms of a Super-Resolution Factor (SRF). We also derive a new Cramér-Rao lower bound for the clumps model, and as a result, implies that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems.

Due to the physical nature of wave propagation and diffraction, there is a fundamental diffraction barrier in optical imaging systems which is called the diffraction limit or resolution limit. Rayleigh investigated this problem and formulated the well-known Rayleigh limit. However, the Rayleigh limit is empirical and only considers the resolving ability of the human visual system. On the other hand, resolving sources separated below the Rayleigh limit to achieve so-called “super-resolution” has been demonstrated in many numerical experiments.

In this talk, we will propose a new concept “computational resolution limit” which reveals the fundamental limits in superresolving the number and locations of point sources from a data-processing point of view. We will quantitatively characterize the computational resolution limits by the signal-to-noise ratio, the sparsity of sources, and the cutoff frequency of the imaging system. As a direct consequence, it is demonstrated that $l_0$ optimization achieves the optimal order resolution in solving super-resolution problems. For the case of resolving two point sources, the resolution estimate is improved to an exact formula, which answers the long-standing question of diffraction limit in a general circumstance. We will also propose an optimal algorithm to distinguish images generated by single or multiple point sources. Generalization of our results to the imaging of positive sources and imaging in multi-dimensional spaces will be briefly discussed as well.

The total (gradient) variation has been used in many imaging applications following the seminal work of Rudin, Osher and Fatemi.[1] In this talk, I will describe a "support stability'' result for total-variation regularized inverse problems: under some assumptions, the solutions at low noise and low regularization have the same number of values as the unknown image, and their level sets converge to those of the unknown image.

It is a joint work with Romain Petit and Yohann De Castro.

[1] L. I. Rudin, S. Osher, E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, Volume 60, Issues 1–4, 1 November 1992, Pages 259-268.

Recent years have seen the development of super-resolution variational optimisation optimisation in measure spaces. These so-called off-the-grid approaches offer both theoretical (uniqueness, reconstruction guarantees) and numerical results, with very convincing results in biomedical imaging. However, the gridless variational optimisation is formulated for reconstruction of point sources, which is not always suitable for biomedical imaging applications: more realistic biological structures such as curves should also be reconstructed, to represent blood vessels or filaments for instance.

We propose a new strategy for the reconstruction of curves in an image through an off-the-grid variational framework, inspired by the reconstruction of spikes in the literature. We introduce a new functional CROC on the space of 2-dimensional Radon measures with finite divergence denoted V. Our main contribution lies in the sharp characterisation of the extreme points of the unit ball of the V -norm: there are exactly measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise characterisation of our functional minimisers and further opening the avenue for the algorithmic implementation.