  We study the ubiquitous super-resolution problem, in which one aims at
localizing positive point sources in an image, blurred by the point spread
function of the imaging device. To recover the point sources, we propose to
solve a convex feasibility program, which simply finds a nonnegative Borel
measure that agrees with the observations collected by the imaging device.
  In the absence of imaging noise, we show that solving this convex program
uniquely retrieves the point sources, provided that the imaging device collects
enough observations. This result holds true if the point spread function of the
imaging device can be decomposed into horizontal and vertical components, and
if the translations of these components form a Chebyshev system, i.e., a system
of continuous functions that loosely behave like algebraic polynomials.
  Building upon recent results for one-dimensional signals [1], we prove that
this super-resolution algorithm is stable, in the generalized Wasserstein
metric, to model mismatch (i.e., when the image is not sparse) and to additive
imaging noise. In particular, the recovery error depends on the noise level and
how well the image can be approximated with well-separated point sources. As an
example, we verify these claims for the important case of a Gaussian point
spread function. The proofs rely on the construction of novel interpolating
polynomials---which are the main technical contribution of this paper---and
partially resolve the question raised in [2] about the extension of the
standard machinery to higher dimensions.
