A mathematical theory of computational resolution limit in one dimension

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.