From Spikes to Curves: Super Resolution via Off-The-Grid Methods

The thesis addresses the problem of super-resolution in fluorescence microscopy. As in any optical system, the resolution of images obtained by a microscope is fundamentally limited by the phenomenon of diffraction. Super-resolution techniques aim to overcome this physical limitation by mathematically exploiting prior knowledge of the acquisition process, typically modelled through a Point Spread Function (PSF). In this thesis, super resolution is reformulated as an inverse problem on the space of Radon measures, focusing on the reconstruction of spikes and curves in an off-the grid setting. Grid-less methods have been proven effective in super-resolution tasks, as they avoid discretizing the domain and thus allow precise reconstruction of the signal’s location. The reconstruction of spikes in a measure-theoretic framework is a well-studied topic. In this context, the Sliding Frank-Wolfe Algorithm - a state-of-the-art method based on the celebrated Conditional Gradient algorithm - is compared with a recently proposed Forward-Backward method adapted to Radon measures. Numerical experiments are conducted on both synthetic and real datasets, with particular attention paid to the reconstruction of closely spaced spikes, in order to evaluate the super-resolution capabilities of the methods. In contrast, the reconstruction of curves represents a more recent topic within the image processing community. Building on previous works, the space of vector Radon measures with finite divergence is adopted as a suitable optimization framework. A detailed analysis is carried out on the corresponding variational problem. The objective functional is shown to be a difference of convex functions, paving the way for the application of the well-known DC Algorithm.