Super-Resolution of Positive Sources on an Arbitrarily Fine Grid

Dynamic Super-Resolution in Particle Tracking Problems

“…He derived both the lower and upper bounds for the minimax error in the amplitude reconstruction, emphasizing the importance of sparsity in the super-resolution. More recently, inspired by the huge success of new super-resolution modalities [8,25,26,45,51] and the popularity of researches for super-resolution algorithms [5,9,18,21,41,42,50], the study of super-resolution capability of imaging problems also becomes popular in applied mathematics. In [16], the authors considered n-sparse point sources supported on a grid (spacing by Δ) and obtained sharper lower and upper bounds for the minimax error of amplitude recovery than those in [20].…”

A note on spike localization for line spectrum estimation

“…The modern convex relaxation approach, optimizing the ℓ 1 , total variation, and atomic norms, has been extensively developed in [5,7,8,27,40,41], to name a few. The most active area of theoretical analysis in line spectrum estimation is super-resolution [1][2][3][4][7][8][9][10][11][13][14][15][16][17][18][19][21][22][23][24][25][26][28][29][30]32,33,37,41], where the goal is to recover the spectrum when the minimal separation is smaller than the Rayleigh distance O (1/K). The first work on superresolution stability was [13], where Donoho introduced the concept of the Rayleigh index and demonstrated the connection between the Rayleigh index, super-resolution factor (SRF), and the allowed perturbation size.…”

“…Among these works, several papers have discussed the recovery of positive spikes. For example, [33], and [32] analyzed individual spike recovery errors in terms of the super-resolution factor under Rayleigh regularity assumptions. The result in [37] shows that the spectrum can be exactly recovered without assuming spectral gaps if the observed signal is a noiseless superposition of certain point spread functions, and in particular, Gaussian point spread functions.…”