Sparse Super-Resolution from Laplace Measurements

We propose a theoretical analysis of the super-resolution performance of the BLASSO “off-the-grid” recovery method from Laplace transform measurements. This transform is not translation invariant, thus requiring the use of theoretical and algorithmic tools that go beyond traditional deconvolution-based methods. We show that the BLASSO offers a stable and computationally tractable super-resolution of positive spikes. In particular, when the signal-to-noise ratio is of the order of 1/t2N−1 (where t is the spacing between the N spikes to recover), the BLASSO program outputs the correct number of spikes. This result suggests that the BLASSO should be a method of choice to tackle challenging Laplace inversions, which are at the heart of recently proposed fluorescence imaging methods.