  Given an image generated by the convolution of point sources with a
band-limited function, the deconvolution problem is to reconstruct the source
number, positions, and amplitudes. This problem arises from many important
applications in imaging and signal processing. It is well-known that it is
impossible to resolve the sources when they are close enough in practice.
Rayleigh investigated this problem and formulated a resolution limit, the
so-called Rayleigh limit, for the case of two sources with identical
amplitudes. On the other hand, many numerical experiments demonstrate that a
stable recovery of the sources is possible even if the sources are separated
below the Rayleigh limit. In this paper, a mathematical theory for the
deconvolution problem in one dimension is developed. The theory addresses the
issue when one can recover the source number exactly from noisy data. The key
is a new concept "computational resolution limit" which is defined to be the
minimum separation distance between the sources such that exact recovery of the
source number is possible. This new resolution limit is determined by the
signal-to-noise ratio and the sparsity of sources, in addition to the cutoff
frequency of the image. Quantitative bounds for this limit is derived, which
reveal the importance of the sparsity as well as the signal-to-noise ratio to
the recovery problem. The stability for recovering the source positions is also
analyzed under a condition on their separation distances. Moreover, a
singular-value-thresholding algorithm is proposed to recover the source number
of a cluster of closely spaced point sources and to verify our theoretical
results on the computational resolution limit. The results are based on a
multipole expansion method and a non-linear approximation theory in Vandermonde
space.
